Free Printable Math Worksheets for Algebra 2

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  • Order of operations
  • Evaluating expressions
  • Simplifying algebraic expressions
  • Multi-step equations
  • Work word problems
  • Distance-rate-time word problems
  • Mixture word problems
  • Absolute value equations
  • Multi-step inequalities
  • Compound inequalities
  • Absolute value inequalities
  • Discrete relations
  • Continuous relations
  • Evaluating and graphing functions
  • Review of linear equations
  • Graphing absolute value functions
  • Graphing linear inequalities
  • Direct and inverse variation
  • Systems of two linear inequalities
  • Systems of two equations
  • Systems of two equations, word problems
  • Points in three dimensions
  • Systems of three equations, elimination
  • Systems of three equations, substitution
  • Basic matrix operations
  • Matrix multiplication
  • All matrix operations combined
  • Matrix inverses
  • Geometric transformations with matrices
  • Operations with complex numbers
  • Properties of complex numbers
  • Rationalizing imaginary denominators
  • Properties of parabolas
  • Vertex form
  • Graphing quadratic inequalities
  • Factoring quadratic expressions
  • Solving quadratic equations w/ square roots
  • Solving quadratic equations by factoring
  • Completing the square
  • Solving equations by completing the square
  • Solving equations with the quadratic formula
  • The discriminant
  • Naming and simple operations
  • Factoring a sum/difference of cubes
  • Factoring by grouping
  • Factoring quadratic form
  • Factoring using all techniques
  • Factors and Zeros
  • The Remainder Theorem
  • Irrational and Imaginary Root Theorems
  • Descartes' Rule of Signs
  • More on factors, zeros, and dividing
  • The Rational Root Theorem
  • Polynomial equations
  • Basic shape of graphs of polynomials
  • Graphing polynomial functions
  • The Binomial Theorem
  • Evaluating functions
  • Function operations
  • Inverse functions
  • Simplifying radicals
  • Operations with radical expressions
  • Dividing radical expressions
  • Radicals and rational exponents
  • Simplifying rational exponents
  • Square root equations
  • Rational exponent equations
  • Graphing radicals
  • Graphing & properties of parabolas
  • Equations of parabolas
  • Graphing & properties of circles
  • Equations of circles
  • Graphing & properties of ellipses
  • Equations of ellipses
  • Graphing & properties of hyperbolas
  • Equations of hyperbolas
  • Classifying conic sections
  • Eccentricity
  • Systems of quadratic equations
  • Graphing simple rational functions
  • Graphing general rational functions
  • Simplifying rational expressions
  • Multiplying / dividing rational expressions
  • Adding / subtracting rational expressions
  • Complex fractions
  • Solving rational equations
  • The meaning of logarithms
  • Properties of logarithms
  • The change of base formula
  • Writing logs in terms of others
  • Logarithmic equations
  • Inverse functions and logarithms
  • Exponential equations not requiring logarithms
  • Exponential equations requiring logarithms
  • Graphing logarithms
  • Graphing exponential functions
  • Discrete exponential growth and decay word problems
  • Continuous exponential growth and decay word problems
  • General sequences
  • Arithmetic sequences
  • Geometric sequences
  • Comparing Arithmetic/Geometric Sequences
  • General series
  • Arithmetic series
  • Arithmetic/Geometric Means w/ Sequences
  • Finite geometric series
  • Infinite geometric series
  • Right triangle trig: Evaluating ratios
  • Right triangle trig: Missing sides/angles
  • Angles and angle measure
  • Co-terminal angles and reference angles
  • Arc length and sector area
  • Trig ratios of general angles
  • Exact trig ratios of important angles
  • The Law of Sines
  • The Law of Cosines
  • Graphing trig functions
  • Translating trig functions
  • Angle Sum/Difference Identities
  • Double-/Half-Angle Identities
  • Sample spaces and The Counting Principle
  • Independent and dependent events
  • Mutualy exclusive events
  • Permutations
  • Combinations
  • Permutations vs combinations
  • Probability using permutations and combinations

Algebra Worksheets

Welcome to the Algebra worksheets page at Math-Drills.com, where unknowns are common and variables are the norm. On this page, you will find Algebra worksheets for middle school students on topics such as algebraic expressions, equations and graphing functions.

This page starts off with some missing numbers worksheets for younger students. We then get right into algebra by helping students recognize and understand the basic language related to algebra. The rest of the page covers some of the main topics you'll encounter in algebra units. Remember that by teaching students algebra, you are helping to create the future financial whizzes, engineers, and scientists that will solve all of our world's problems.

Algebra is much more interesting when things are more real. Solving linear equations is much more fun with a two pan balance, some mystery bags and a bunch of jelly beans. Algebra tiles are used by many teachers to help students understand a variety of algebra topics. And there is nothing like a set of co-ordinate axes to solve systems of linear equations.

Most Popular Algebra Worksheets this Week

Combining Like Terms and Solving Simple Linear Equations

Algebraic Properties, Rules and Laws Worksheets

algebra worksheet 2 5 answer key

The commutative law or commutative property states that you can change the order of the numbers in an arithmetic problem and still get the same results. In the context of arithmetic, it only works with addition or multiplication operations , but not mixed addition and multiplication. For example, 3 + 5 = 5 + 3 and 9 × 5 = 5 × 9. A fun activity that you can use in the classroom is to brainstorm non-numerical things from everyday life that are commutative and non-commutative. Putting on socks, for example, is commutative because you can put on the right sock then the left sock or you can put on the left sock then the right sock and you will end up with the same result. Putting on underwear and pants, however, is non-commutative.

  • The Commutative Law Worksheets The Commutative Law of Addition (Numbers Only) The Commutative Law of Addition (Some Variables) The Commutative Law of Multiplication (Numbers Only) The Commutative Law of Multiplication (Some Variables)

The associative law or associative property allows you to change the grouping of the operations in an arithmetic problem with two or more steps without changing the result. The order of the numbers stays the same in the associative law. As with the commutative law, it applies to addition-only or multiplication-only problems. It is best thought of in the context of order of operations as it requires that parentheses must be dealt with first. An example of the associative law is: (9 + 5) + 6 = 9 + (5 + 6). In this case, it doesn't matter if you add 9 + 5 first or 5 + 6 first, you will end up with the same result. Students might think of some examples from their experience such as putting items on a tray at lunch. They could put the milk and vegetables on their tray first then the sandwich or they could start with the vegetables and sandwich then put on the milk. If their tray looks the same both times, they will have modeled the associative law. Reading a book could be argued as either associative or nonassociative as one could potentially read the final chapters first and still understand the book as well as someone who read the book the normal way.

  • The Associative Law Worksheets The Associative Law of Addition (Whole Numbers Only) The Associative Law of Multiplication (Whole Numbers Only)

Inverse relationships worksheets cover a pre-algebra skill meant to help students understand the relationship between multiplication and division and the relationship between addition and subtraction.

  • Inverse Mathematical Relationships with One Blank Addition and Subtraction Easy Addition and Subtraction Harder All Multiplication and Division Facts 1 to 18 in color (no blanks) Multiplication and Division Range 1 to 9 Multiplication and Division Range 5 to 12 Multiplication and Division All Inverse Relationships Range 2 to 9 Multiplication and Division All Inverse Relationships Range 5 to 12 Multiplication and Division All Inverse Relationships Range 10 to 25
  • Inverse Mathematical Relationships with Two Blanks Addition and Subtraction (Sums 1-18) Addition and Subtraction Inverse Relationships with 1 Addition and Subtraction Inverse Relationships with 2 Addition and Subtraction Inverse Relationships with 3 Addition and Subtraction Inverse Relationships with 4 Addition and Subtraction Inverse Relationships with 5 Addition and Subtraction Inverse Relationships with 6 Addition and Subtraction Inverse Relationships with 7 Addition and Subtraction Inverse Relationships with 8 Addition and Subtraction Inverse Relationships with 9 Addition and Subtraction Inverse Relationships with 10 Addition and Subtraction Inverse Relationships with 11 Addition and Subtraction Inverse Relationships with 12 Addition and Subtraction Inverse Relationships with 13 Addition and Subtraction Inverse Relationships with 14 Addition and Subtraction Inverse Relationships with 15 Addition and Subtraction Inverse Relationships with 16 Addition and Subtraction Inverse Relationships with 17 Addition and Subtraction Inverse Relationships with 18

The distributive property is an important skill to have in algebra. In simple terms, it means that you can split one of the factors in multiplication into addends, multiply each addend separately, add the results, and you will end up with the same answer. It is also useful in mental math, an example of which should help illustrate the definition. Consider the question, 35 × 12. Splitting the 12 into 10 + 2 gives us an opportunity to complete the question mentally using the distributive property. First multiply 35 × 10 to get 350. Second, multiply 35 × 2 to get 70. Lastly, add 350 + 70 to get 420. In algebra, the distributive property becomes useful in cases where one cannot easily add the other factor before multiplying. For example, in the expression, 3(x + 5), x + 5 cannot be added without knowing the value of x. Instead, the distributive property can be used to multiply 3 × x and 3 × 5 to get 3x + 15.

  • Distributive Property Worksheets Distributive Property (Answers do not include exponents) Distributive Property (Some answers include exponents) Distributive Property (All answers include exponents)

Students should be able to substitute known values in for an unknown(s) in an expression and evaluate the expression's value.

  • Evaluating Expressions with Known Values Evaluating Expressions with One Variable, One Step and No Exponents Evaluating Expressions with One Variable and One Step Evaluating Expressions with One Variable and Two Steps Evaluating Expressions with Up to Two Variables and Two Steps Evaluating Expressions with Up to Two Variables and Three Steps Evaluating Expressions with Up to Three Variables and Four Steps Evaluating Expressions with Up to Three Variables and Five Steps

As the title says, these worksheets include only basic exponent rules questions. Each question only has two exponents to deal with; complicated mixed up terms and things that a more advanced student might work out are left alone. For example, 4 2 is (2 2 ) 2 = 2 4 , but these worksheets just leave it as 4 2 , so students can focus on learning how to multiply and divide exponents more or less in isolation.

  • Exponent Rules for Multiplying, Dividing and Powers Mixed Exponent Rules (All Positive) Mixed Exponent Rules (With Negatives) Multiplying Exponents (All Positive) Multiplying Exponents (With Negatives) Multiplying the Same Exponent with Different Bases (All Positive) Multiplying the Same Exponent with Different Bases (With Negatives) Dividing Exponents with a Greater Exponent in Dividend (All Positive) Dividing Exponents with a Greater Exponent in Dividend (With Negatives) Dividing Exponents with a Greater Exponent in Divisor (All Positive) Dividing Exponents with a Greater Exponent in Divisor (With Negatives) Powers of Exponents (All Positive) Powers of Exponents (With Negatives)

Knowing the language of algebra can help to extract meaning from word problems and to situations outside of school. In these worksheets, students are challenged to convert phrases into algebraic expressions.

  • Translating Algebraic Phrases into Expressions Translating Algebraic Phrases into Expressions (Simple Version) Translating Algebraic Phrases into Expressions (Complex Version)

Combining like terms is something that happens a lot in algebra. Students can be introduced to the topic and practice a bit with these worksheets. The bar is raised with the adding and subtracting versions that introduce parentheses into the expressions. For students who have a good grasp of fractions, simplifying simple algebraic fractions worksheets present a bit of a challenge over the other worksheets in this section.

  • Simplifying Expressions by Combining Like Terms Simplifying Linear Expressions with 3 terms Simplifying Linear Expressions with 4 terms Simplifying Linear Expressions with 5 terms Simplifying Linear Expressions with 6 to 10 terms
  • Simplifying Expressions by Combining Like Terms with Some Arithmetic Adding and simplifying linear expressions Adding and simplifying linear expressions with multipliers Adding and simplifying linear expressions with some multipliers . Subtracting and simplifying linear expressions Subtracting and simplifying linear expressions with multipliers Subtracting and simplifying linear expressions with some multipliers Mixed adding and subtracting and simplifying linear expressions Mixed adding and subtracting and simplifying linear expressions with multipliers Mixed adding and subtracting and simplifying linear expressions with some multipliers Simplify simple algebraic fractions (easier) Simplify simple algebraic fractions (harder)
  • Rewriting Linear Equations Rewrite Linear Equations in Standard Form Convert Linear Equations from Standard to Slope-Intercept Form Convert Linear Equations from Slope-Intercept to Standard Form Convert Linear Equations Between Standard and Slope-Intercept Form
  • Rewriting Formulas Rewriting Formulas (addition and subtraction; about one step) Rewriting Formulas (addition and subtraction; about two steps) Rewriting Formulas ( multiplication and division ; about one step)

Linear Expressions and Equations

algebra worksheet 2 5 answer key

In these worksheets, the unknown is limited to the question side of the equation which could be on the left or the right of equal sign.

  • Missing Numbers in Equations with Blanks as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Blanks in Any Position )
  • Missing Numbers in Equations with Symbols as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Symbols in Any Position )
  • Solving Equations with Addition and Symbols as Unknowns Equalities with Addition (0 to 9) Symbol Unknowns Equalities with Addition (1 to 12) Symbol Unknowns Equalities with Addition (1 to 15) Symbol Unknowns Equalities with Addition (1 to 25) Symbol Unknowns Equalities with Addition (1 to 99) Symbol Unknowns
  • Missing Numbers in Equations with Variables as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Variables Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Variables in Any Position )
  • Solving Simple Linear Equations Solving Simple Linear Equations with Values from -9 to 9 (Unknown on Left Side) Solving Simple Linear Equations with Values from -99 to 99 (Unknown on Left Side) Solving Simple Linear Equations with Values from -9 to 9 (Unknown on Right or Left Side) Solving Simple Linear Equations with Values from -99 to 99 (Unknown on Right or Left Side)
  • Determining Linear Equations from Slopes, y-intercepts and Points Determine a Linear Equation from the Slope and y-intercept Determine a Linear Equation from the Slope and a Point Determine a Linear Equation from Two Points Determine a Linear Equation from Two Points by Graphing

Graphing linear equations and reading existing graphs give students a visual representation that is very useful in understanding the concepts of slope and y-intercept.

  • Graphing Linear Equations Graph Slope-Intercept Equations
  • Determinging Linear Equations from Graphs Determine the Equation from a Graph Determine the Slope from a Graph Determine the y-intercept from a Graph Determine the x-intercept from a Graph Determine the slope and y-intercept from a Graph Determine the slope and intercepts from a Graph Determine the slope, intercepts and equation from a Graph

Solving linear equations with jelly beans is a fun activity to try with students first learning algebraic concepts. Ideally, you will want some opaque bags with no mass, but since that isn't quite possible (the no mass part), there is a bit of a condition here that will actually help students understand equations better. Any bags that you use have to be balanced on the other side of the equation with empty ones.

Probably the best way to illustrate this is through an example. Let's use 3 x + 2 = 14. You may recognize the x as the unknown which is actually the number of jelly beans we put in each opaque bag. The 3 in the 3 x means that we need three bags. It's best to fill the bags with the required number of jelly beans out of view of the students, so they actually have to solve the equation.

On one side of the two-pan balance, place the three bags with x jelly beans in each one and two loose jelly beans to represent the + 2 part of the equation. On the other side of the balance, place 14 jelly beans and three empty bags which you will note are required to "balance" the equation properly. Now comes the fun part... if students remove the two loose jelly beans from one side of the equation, things become unbalanced, so they need to remove two jelly beans from the other side of the balance to keep things even. Eating the jelly beans is optional. The goal is to isolate the bags on one side of the balance without any loose jelly beans while still balancing the equation.

The last step is to divide the loose jelly beans on one side of the equation into the same number of groups as there are bags. This will probably give you a good indication of how many jelly beans there are in each bag. If not, eat some and try again. Now, we realize this won't work for every linear equation as it is hard to have negative jelly beans, but it is another teaching strategy that you can use for algebra.

Despite all appearances, equations of the type a/ x are not linear. Instead, they belong to a different kind of equations. They are good for combining them with linear equations, since they introduce the concept of valid and invalid answers for an equation (what will be later called the domain of a function). In this case, the invalid answers for equations in the form a/ x , are those that make the denominator become 0.

  • Solving Linear Equations Combining Like Terms and Solving Simple Linear Equations Solving a x = c Linear Equations Solving a x = c Linear Equations including negatives Solving x /a = c Linear Equations Solving x /a = c Linear Equations including negatives Solving a/ x = c Linear Equations Solving a/ x = c Linear Equations including negatives Solving a x + b = c Linear Equations Solving a x + b = c Linear Equations including negatives Solving a x - b = c Linear Equations Solving a x - b = c Linear Equations including negatives Solving a x ± b = c Linear Equations Solving a x ± b = c Linear Equations including negatives Solving x /a ± b = c Linear Equations Solving x /a ± b = c Linear Equations including negatives Solving a/ x ± b = c Linear Equations Solving a/ x ± b = c Linear Equations including negatives Solving various a/ x ± b = c and x /a ± b = c Linear Equations Solving various a/ x ± b = c and x /a ± b = c Linear Equations including negatives Solving linear equations of all types Solving linear equations of all types including negatives

Linear Systems

algebra worksheet 2 5 answer key

  • Solving Systems of Linear Equations Easy Linear Systems with Two Variables Easy Linear Systems with Two Variables including negative values Linear Systems with Two Variables Linear Systems with Two Variables including negative values Easy Linear Systems with Three Variables; Easy Easy Linear Systems with Three Variables including negative values Linear Systems with Three Variables Linear Systems with Three Variables including negative values
  • Solving Systems of Linear Equations by Graphing Solve Linear Systems by Graphing (Solutions in first quadrant only) Solve Standard Linear Systems by Graphing Solve Slope-Intercept Linear Systems by Graphing Solve Various Linear Systems by Graphing Identify the Dependent Linear System by Graphing Identify the Inconsistent Linear System by Graphing

Quadratic Expressions and Equations

algebra worksheet 2 5 answer key

  • Simplifying (Combining Like Terms) Quadratic Expressions Simplifying quadratic expressions with 5 terms Simplifying quadratic expressions with 6 terms Simplifying quadratic expressions with 7 terms Simplifying quadratic expressions with 8 terms Simplifying quadratic expressions with 9 terms Simplifying quadratic expressions with 10 terms Simplifying quadratic expressions with 5 to 10 terms
  • Adding/Subtracting and Simplifying Quadratic Expressions Adding and simplifying quadratic expressions. Adding and simplifying quadratic expressions with multipliers. Adding and simplifying quadratic expressions with some multipliers. Subtracting and simplifying quadratic expressions. Subtracting and simplifying quadratic expressions with multipliers. Subtracting and simplifying quadratic expressions with some multipliers. Mixed adding and subtracting and simplifying quadratic expressions. Mixed adding and subtracting and simplifying quadratic expressions with multipliers. Mixed adding and subtracting and simplifying quadratic expressions with some multipliers.
  • Multiplying Factors to Get Quadratic Expressions Multiplying Factors of Quadratics with Coefficients of 1 Multiplying Factors of Quadratics with Coefficients of 1 or -1 Multiplying Factors of Quadratics with Coefficients of 1, or 2 Multiplying Factors of Quadratics with Coefficients of 1, -1, 2 or -2 Multiplying Factors of Quadratics with Coefficients up to 9 Multiplying Factors of Quadratics with Coefficients between -9 and 9

The factoring quadratic expressions worksheets in this section provide many practice questions for students to hone their factoring strategies. If you would rather worksheets with quadratic equations, please see the next section. These worksheets come in a variety of levels with the easier ones are at the beginning. The 'a' coefficients referred to below are the coefficients of the x 2 term as in the general quadratic expression: ax 2 + bx + c. There are also worksheets in this section for calculating sum and product and for determining the operands for sum and product pairs.

  • Factoring Quadratic Expressions Factoring Quadratic Expressions with Positive 'a' coefficients of 1 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients of 1 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients of 1 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 4 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 4 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 4 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 5 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 5 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 5 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 9 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 9 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 9 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 81 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 81 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 81 with a Common Factor Step Calculating Sum and Product (Operand Range 0 to 9 ) ✎ Calculating Sum and Product (Operand Range 1 to 9 ) ✎ Calculating Sum and Product (Operand Range 0 to 9 Including Negatives ) ✎ Calculating Sum and Product (Operand Range 1 to 9 Including Negatives ) ✎ Calculating Sum and Product (Operand Range -20 to 20 ) ✎ Calculating Sum and Product (Operand Range -99 to 99 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 9 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 9 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 12 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 12 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 9 Including Negatives ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 9 Including Negatives ) ✎ Determining Operands from Sum and Product Pairs (Operand Range -20 to 20 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range -99 to 99 ) ✎

Whether you use trial and error, completing the square or the general quadratic formula, these worksheets include a plethora of practice questions with answers. In the first section, the worksheets include questions where the quadratic expressions equal 0. This makes the process similar to factoring quadratic expressions, with the additional step of finding the values for x when the expression is equal to 0. In the second section, the expressions are generally equal to something other than x, so there is an additional step at the beginning to make the quadratic expression equal zero.

  • Solving Quadratic Equations that Equal Zero Solving Quadratic Equations with Positive 'a' coefficients of 1 Solving Quadratic Equations with Positive or Negative 'a' coefficients of 1 Solving Quadratic Equations with Positive or Negative 'a' coefficients of 1 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 4 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 4 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 4 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 5 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 5 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 5 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 9 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 9 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 9 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 81 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 81 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 81 with a Common Factor Step
  • Solving Quadratic Equations that Equal an Integer Solving Quadratic Equations for x ("a" coefficients of 1) Solving Quadratic Equations for x ("a" coefficients of 1 or -1) Solving Quadratic Equations for x ("a" coefficients up to 4) Solving Quadratic Equations for x ("a" coefficients between -4 and 4) Solving Quadratic Equations for x ("a" coefficients up to 81) Solving Quadratic Equations for x ("a" coefficients between -81 and 81)

Other Polynomial and Monomial Expressions & Equations

algebra worksheet 2 5 answer key

  • Simplifying Polynomials That Involve Addition And Subtraction Addition and Subtraction; 1 variable; 3 terms Addition and Subtraction; 1 variable; 4 terms Addition and Subtraction; 2 variables; 4 terms Addition and Subtraction; 2 variables; 5 terms Addition and Subtraction; 2 variables; 6 terms
  • Simplifying Polynomials That Involve Multiplication And Division Multiplication and Division; 1 variable; 3 terms Multiplication and Division; 1 variable; 4 terms Multiplication and Division; 2 variables; 4 terms Multiplication and Division; 2 variables; 5 terms
  • Simplifying Polynomials That Involve Addition, Subtraction, Multiplication And Division All Operations; 1 variable; 3 terms All Operations; 1 variable; 4 terms All Operations; 2 variables; 4 terms All Operations; 2 variables; 5 terms All Operations (Challenge)
  • Factoring Expressions That Do Not Include A Squared Variable Factoring Non-Quadratic Expressions with No Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Compound Coefficients, and Negative and Positive Multipliers
  • Factoring Expressions That Always Include A Squared Variable Factoring Non-Quadratic Expressions with All Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Compound Coefficients, and Negative and Positive Multipliers
  • Factoring Expressions That Sometimes Include Squared Variables Factoring Non-Quadratic Expressions with Some Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Compound Coefficients, and Negative and Positive Multipliers
  • Multiplying Polynomials With Two Factors Multiplying a monomial by a binomial Multiplying two binomials Multiplying a monomial by a trinomial Multiplying a binomial by a trinomial Multiplying two trinomials Multiplying two random mon/polynomials
  • Multiplying Polynomials With Three Factors Multiplying a monomial by two binomials Multiplying three binomials Multiplying two binomials by a trinomial Multiplying a binomial by two trinomials Multiplying three trinomials Multiplying three random mon/polynomials

Inequalities

algebra worksheet 2 5 answer key

  • Writing The Inequality That Matches The Graph Writing Inequalities for Graphs
  • Graphing Inequalities On Number Lines Graphing Inequalities (Basic)
  • Solving Linear Inequalities Solving Inequalities Including a Third Term Solving Inequalities Including a Third Term and Multiplication Solving Inequalities Including a Third Term, Multiplication and Division

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  • 2-sided equations

Algebra: 2 sided equations

Gathering variables and constants.

In these algebra worksheets, variables appear on both sides of the equal sign . Students have to simplify and determine the value of the variable.

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Algebra 2 Worksheets: FREE & Printable

Looking for free printable Algebra 2 worksheets and exercises to help you or your students prepare for the Algebra 2 test?

Algebra 2 Worksheets: FREE & Printable

Want Algebra 2 practice questions and activities to measure your exam readiness? Need comprehensive Algebra 2 worksheets to help your students learn Algebra 2 concepts and topics? If so, then look no further.

Here is a perfect and comprehensive collection of FREE Algebra 2 worksheets that would help you or your students in Algebra 2 preparation and practice.

Download our free Mathematics worksheets for the Algebra 2 test.

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IMPORTANT: COPYRIGHT TERMS: Worksheets may not be uploaded to the internet, including classroom/personal websites or network drives. You can download the worksheets and print as many as you need. You can distribute the printed copies to your students, teachers, tutors, and friends. 

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The Absolute Best Book to Ace the Algebra 2 Test

Algebra II for Beginners The Ultimate Step by Step Guide to Acing Algebra II

Algebra 2 concepts, fundamentals and building blocks.

  • Order of Operations
  • Scientific Notation
  • Exponents Operations
  • Evaluating Expressions
  • Simplifying Algebraic Expressions

Equations and Inequalities

  • One–Step Equations
  • Multi–Step Equations
  • Graphing Single–Variable Inequalities
  • One–Step Inequalities
  • Multi-Step Inequalities

System of Equations and Quadratic

  • Systems of Equations
  • Systems of Equations Word Problems
  • Systems of 3 Variable Equations
  • Solving a Quadratic Equation
  • Quadratic Formula and the Discriminant
  • Solving Quadratic Inequalities
  • Graphing Quadratic Functions

Complex Numbers

  • Adding and Subtracting Complex Numbers
  • Multiplying and Dividing Complex Numbers
  • Graphing Complex Numbers
  • Rationalizing Imaginary Denominators
  • Adding and Subtracting Matrices
  • Matrix Multiplication
  • Finding Determinants of a Matrix
  • Finding Inverse of a Matrix
  • Matrix Equations

Polynomial Operations

  • Writing Polynomials in Standard Form
  • Simplifying Polynomial Expressions
  • Adding and Subtracting Polynomials
  • Multiplying Monomials
  • Multiplying and Dividing Monomials
  • Multiplying a Polynomial and a Monomial
  • Multiplying Binomials
  • Factoring Trinomials
  • Operations with Polynomials

Functions Operations

  • Evaluating Function
  • Adding and Subtracting Functions
  • Multiplying and Dividing Functions
  • Composition of Functions

The Most Comprehensive Book to Ace the Algebra 2 Test

Algebra II Practice Workbook The Most Comprehensive Review of Algebra 2

  • Rewriting Logarithms
  • Evalu ating Logarithms
  • Properties of Logarithms
  • Natural Logarithms
  • Exponential Equations and Logarithms
  • Solving Logarithmic Equations

Radical Expressions

  • Simplifying Radical Expressions
  • Multiplying Radical Expressions
  • Simplifying Radical Expressions Involving Fractions
  • Adding and Subtracting Radical Expressions
  • Domain and Range of Radical Functions
  • Solving Radical Equations

Rational Expressions

  • Simplifying and Graphing Rational Expressions
  • Adding and Subtracting Rational Expressions
  • Multiplying and Dividing Rational Expressions
  • Solving Rational Equations and Complex Fractions

Conic Sections

  • Finding the Equation of a Parabola
  • Finding the Focus, Vertex, and Directrix of a Parabola
  • Writing the Equation of a Hyperbola in Standard Form
  • Classifying a Conic Section (in Standard Form)

Trigonometric Functions

  • Trig Ratios of General Angles
  • Sketch Each Angle in Standard Position
  • Finding Co-Terminal Angles and Reference Angles
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Algebra 2 Practice Questions

Fractions and decimals, real numbers and integers, proportions and ratios, algebraic expressions, linear functions, polynomials, exponents and radicals, solid figures.

Looking for the best resource to help you succeed on the Algebra test?

The Best Books to Ace the Algebra Test

College Algebra Practice Workbook The Most Comprehensive Review of College Algebra

High school algebra i a comprehensive review and step-by-step guide to mastering high school algebra 1, 10 full length clep college algebra practice tests the practice you need to ace the clep college algebra test.

by: Effortless Math Team about 3 years ago (category: Blog , Free Math Worksheets )

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Algebra i practice workbook the most comprehensive review of algebra 1, the ultimate algebra bundle from pre-algebra to algebra ii.

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Algebra 2 Online Lessons & Videos

Algebra 2 lesson and worksheet pdfs, course syllabus, --- chapter 1 lessons ---, 1.1 real numbers, 1.2 simplifying algebraic expressions & equations, 1.3 solving linear equations, 1.4 solving linear inequalities, 1.5 quadratic equations, 1.6 interpreting word problems, --- chapter 1 homework & worksheets ---.

1.1 Homework: (Click for video) Pg 6: Problem Set 1, 2, 7, 9-11 Pg 9: Problem Set 10, 11, 12, 14
1.2 Homework: (Click for video) Pg 13: Problem Set 4-6 Pg 16: Problem Set 1-3, 6, 9, 10 Pg 20: Problem Set 7, 8
1.3 Homework: (Click for video) Pg 23: Problem Set 4, 12 Pg 26: Problem Set 2, 13 Pg 30: Problem Set 1, 2, 5, 12
1.4 Homework: (Click for video) Pg 34: Problem Set 1, 4 Pg 38: Problem Set 7, 9 Pg 41: Problem Set 7, 11, 13
1.5 Homework: (Click for video) Pg 47: Problem Set 4, 7, 11, 14 Pg 50: Problem Set 4, 6, 10, 12
1.6 Homework: (Click for video) Worksheet
Chapter 1 Review Review Key

--- Chapter 2 Lessons ---

2.1 finding the slope & equation of a line, 2.2 standard form of a line, 2.3 graphing lines, 2.4 relations & functions, 2.5 graphing linear inequalities of two variables, 2.6 graphing absolute value functions, --- chapter 2 homework & worksheets ---.

2.1 Homework: Pg 63: 1-4 Pg 64: 7, 8 Pg 68: 1-3, 9, 10
2.2 Homework: Pg 73: Problem Set 11, 13 Pg 76: Problem Set 1, 3, 6 Pg 80: Problem Set 5, 10, 11
2.3 Homework: Pg 85: Problem Set 1, 6, 13 Pg 88: Problem Set 1, 2, 8, 10
2.4 Homework: Pg 93: Problem Set 1, 2, 5, 6 Pg 94: Problem Set 8-10, 13 Pg 101: Problem Set 1, 2 Pg 102: Problem Set 7, 9
2.5 Homework: Pg 106: 1, 2 Pg 112: 1, 2, 5-7
2.6 Homework: Pg 124-125: Problem Set 5-10
Chapter 2 Test Review Answer Key (Contains all answers except #s 26-28 & 35-41) Answers for 26-28, 35-41

--- Chapter 3 Lessons ---

3.1 solving linear systems by graphing, 3.2 solving linear systems by substitution, 3.3 solving linear systems by elimination, 3.4 graphing and solving linear inequalities, 3.5 solving linear systems in three variables, --- chapter 3 homework & worksheets ---.

3.1 Homework: Pg 163: 5-9 Pg 164: 11
3.2 Homework: Pg 177-178: Problem Set 1-4 Pg 181: Problem Set 5
3.3 Homework: Pg 188: Problem Set 1, 2, 5 Pg 201-202: Problem Set 3, 4, 6, 12
3.4 Homework: Pg 220: Guided Practice 1-4
3.5 Homework: None, See Review for Problems
Word Problems Worksheet Solutions Video
Chapter 3 Review Pt 1 Answer Key
Chapter 3 Review Pt 2 Answer Key

--- Chapter 4 Lessons ---

4.1 operations of matrices, 4.2 matrix multiplication, 4.3 determinants, 4.4 identity and inverse matrices, 4.5 solving linear systems using matrices, --- chapter 4 homework & worksheets ---.

4.1 Homework: Pg 242: Problem Set 1, 3, 4, 10, 11 Pg 246: Problem Set 2, 6
4.2 Homework: Pg 250: Problem Set 1, 2, 6, 7, 8, 11
4.3 Homework: Pg 264: Problem Set 5-8
4.4 Homework: Pg 274-275: Problem Set 2-7
4.5 Homework: Pg 286: 1-3 Pg 291-292: 10-12
Chapter 4 Project: "The Sandwich Project" 2017-2018 Ingredients by Hour
Chapter 3 & 4 Review Material: Test Review Review Key Word Document

--- Chapter 5 Lessons ---

5.1 factoring quadratics:.

Day 1: FOILing and Basic Factoring
Day 2: Slide & Divide Factoring and Squaring a Binomial
Day 3: Solving Quadratics by Factoring

5.2 Solving Quadratics Using Square Roots

5.3 complex numbers:.

Day 1: The Complex Unit i and Basic Complex Operations
Day 2: Complex Conjugates and Dividing Complex Numbers
Day 3: Solving Quadratics with Complex Solutions

5.4 Completing the Square:

Day 1: Completing the Square when a = 1
Day 2: Completing the Square when a =/= 1

5.5 The Quadratic Formula

5.6 graphs of quadratic functions:.

Day 1: Vertex Form and Transformations
Day 2: Standard Form and Finding the Vertex

--- Chapter 5 Homework & Worksheets ---

5.1 Day 1 Homework: Worksheet Worksheet Solutions
5.1 Day 2 Homework: Pg 303: Problem Set 4-8, 11, 13
5.1 Day 3 Homework: Pg 311: Problem Set 1-4, 6, 8
5.2 Homework: Pg 315-316: Problem Set 4, 7, 8 Pg 321: Problem Set 2, 3, 9, 10
5.3 Day 1 Homework: Pg 325: Problem Set 1, 2, 7, 13, 14 Pg 329: Problem Set 2-4
5.3 Day 2 Homework: Pg 329: Problem Set 9, 10, 12-14
5.3 Day 3 Homework: Pg 331: Problem Set 1-7
5.4 Day 1 Homework: Pg 335: Problem Set 1-3 Pg 336: Problem Set 7-10 Written Solutions
5.4 Day 2 Homework: Pg 339: 1-4, 6 Written Solutions
5.5 Homework: Pg 344-45: Problem Set 1-3 Written Solutions
5.6 Day 1 Homework: Worksheet Solutions Video
5.6 Day 2 Homework: Pg 361: Problem Set 1, 2, 4 (Only find the Vertex) Pg 361: Problem Set 5, 6, 9 (Find Vertex & Sketch Graph)
Chapter 5 Test Review: Review Review Key

--- Chapter 6 Lessons ---

6.1 properties of exponents, 6.2 basic polynomial operations, 6.3 factoring and solving polynomials, 6.4 & 6.5 synthetic division & end behavior, extra ch. 6 lessons:.

Lesson 1: Zeros, Multiplicity & Graphing
Lesson 2: Solving Polynomial Inequalities
Lesson 3: Rational Root Theorem

--- Chapter 6 Homework & Worksheets ---

6.1 Homework: Pg 373 Problem Set: 1, 2, 4, 6, 7, 9
6.2 Homework: Pg 379 Problem Set: 9, 10, 13 (After doing the problem, state the degree & leading coefficient) Pg 382 Problem Set: 3, 7, 9
6.3 Day 1 Homework: Pg 388-389 Problem Set: 1-4, 6, 8
6.3 Day 2 Homework: Pg 389 Problem Set: 10-12 Pg 391 Problem Set: 10-12
6.3 Day 3 Homework: Pg 394: 1-3, 5, 6, 8
6.4 & 6.5 Homework: Pg 402 Problem Set: 1, 2, 4-6
Chapter 6 Review Answers
Extra Ch. 6 Lessons Homework: Lesson #1 Worksheet Answers Lesson #2 Worksheet Answers
Extra Ch. 6 Lessons Review: Worksheet Answers

--- Chapter 7 Lessons ---

7.4 day 1: function operations & composition, 7.1 higher roots & rational exponents, 7.2 graphs and trans. of sq. root functions, 7.3 solving radical functions, 7.4 function operations:.

Day 1: Function Operations & Composition (Covered before 7.1)
Day 2: Function Inverses

--- Chapter 7 Homework and Worksheets ---

7.4 Day 1 Homework: Worksheet PDF Worksheet Key
7.1 Homework: Pg 433 Problem Set: 1-9. On #s 4-6 you do Not have to evaluate, just convert.
7.2 Homework: Pg 443: 2, 3, 9, 10 (On 9 & 10 only state transformations. Do not graph) Pg 448: 3, 4, 6, 11 (on 6 & 11 only state transformations, Do not graph)
7.3 Homework: Pg 459: 3, 7, 8 Pg 462: 1, 2 Pg 464: 1, 3
7.4 Day 2 Homework: Pg 476 Problem Set: 1-3, 5, 6, 9, 11

--- Chapter 8 Lessons ---

8.1 exponential functions.

Day 1: Introduction & Transformations
Day 2: Graphing and Interest Rates

8.2 Logarithm Functions

Day 1: Introduction, Converting & Evaluating
Day 2: Transformations & Graphing

8.3 Properties of Logarithms

8.4 solving exponential and logarithmic equations, --- chapter 8 homework & worksheets ---, 8.1 day 1 homework:, 8.1 day 2 homework:, 8.2 day 1 homework:, 8.2 day 2 homework:, 8.3 homework:, 8.4 homework:, --- chapter 9 lessons ---, 9.1 direct variation, 9.2 graphing rational functions.

Day 1: Identifying x/y-int & V. Asymptotes
Day 2: H. Asymptotes and Graphing

9.3 Simplifying, Multiplying & Dividing Rationals

Day 1: Simplifying by Factoring & Canceling
Day 2: Multiplying & Dividing Multiple Expressions

9.4 Adding & Subtracting Rationals

Day 1: Finding a Common Denominator
Day 2: Factoring Before Adding/Subtracting

9.5 Solving Rational Equations

--- chapter 9 homework & worksheets ---, 9.1 homework:, 9.2 day 1 homework:, 9.2 day 2 homework:, 9.3 day 1 homework:, 9.3 day 2 homework:, 9.4 day 1 homework:, 9.4 day 2 homework:, 9.5 homework:, --- chapter 10 lessons ---, 10.1 distance, midpoint, and parabolas, 10.2 cicrcles, 10.3 ellipses, 10.4 hyperbolas, --- chapter 10 homework & worksheets ---, 10.1 homework:, 10.2 homework:, 10.3 homework:, 10.4 homework:, --- chapter 11 lessons ---, 11.1 introduction to sequences & series.

Day 1: Term Notation & Definitions
Day 2: Writing General & Recursive Rules, Sigma Notation

11.2 Arithmetic Sequences

11.3 geometric sequences, 11.4 infinite series, --- chapter 11 homework & worksheets ---, 11.1 day 1 homework:, 11.1 day 2 homework:, 11.2 homework:, 11.3 homework:, 11.4 homework:, --- chapter 12 lessons ---, 12.1 & 12.2 fundamental counting principle, permutations & combinations, --- chapter 12 homework & worksheets ---, 12.1 & 12.2 homework:, --- final exam review ---, semester 1 final, semester 2 final.

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Basic Algebra Worksheets

Welcome to the Math Salamanders' Basic Algebra Worksheets. Here you will find a range of algebra worksheets to help you learn about basic algebra, including generating and calculating algebraic expressions and solving simple problems.

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Basic Algebra Support

Generate the Expression Worksheets

Calculate the Expression Worksheets

  • Solve the Equations Worksheets
  • More recommended resources
  • Basic Algebra Online Quiz

Want to gain a basic understanding of algebra?

Looking for some simple algebra worksheets?

Do you need a bank of useful algebra resources?

Look no further! The pages you need are below!

Here is our selection of basic algebra sheets to try.

We have split the worksheets up into 3 different sections:

  • Generate the algebra - and write your own algebraic expressions;
  • Calculate the algebra - work out the value of different expressions;
  • Solve the algebra - find the value of the term in the equation.

By splitting the algebra up into sections, you only need to concentrate on one aspect at a time!

Each question sheet comes with its own separate answer sheet.

Want to test yourself to see how well you have understood this skill?.

  • Try our NEW quick quiz at the bottom of this page.

What is an algebraic expression?

An expression is a mathematical statement where variables and operations are combined.

  • 2a + 5 is an expression involving the variable a
  • 5(y 2 - 6) is another expression

What is an algebraic equation?

An equation is where an algebraic expression is equal to something, which might be a number, or another algebraic expression.

  • 2a + 5 = 7 is an equation
  • 5(y 2 - 6) = 3y + 8 is another equation

How to Generate an Expression

When we are generating an expression, we are taking a rule and turning it into algebra.

  • Subtract 6 from n could be written as n - 6.
  • Multiply d by 4 could be written as d x 4 or 4d.
  • Add 5 to p and then double the result is written as (p + 5) x 2 or 2(p + 5)

How to Calculate an Expression

When we are calculating the value of an expression, we work out the value of the expression when we give a value to the variable.

  • p + 5 has a value of 11 when p = 6 because 6 + 5 = 11
  • 4(n - 2) has a value of 32 when n = 10 because 4 x (10 - 2) = 4 x 8 = 32
  • 4(n - 2) has a value of -8 when n = 0 because 4 x (0 - 2) = 4 x (-2) = -8

How to Solve a Simple Equation

When we are solving an equation, we are finding out the value(s) of the variable in the equation.

  • If p + 5 = 9 then p = 4 because 4 + 5 = 9
  • then (n - 2) = 28 ÷ 4 = 7
  • if (n - 2) = 7 then n = 7 + 2 = 9
  • Answer: n = 9
  • means that 3f = 12
  • so f = 12 ÷ 3 = 4
  • Answer: f = 4

Basic Algebra Worksheets for kids

  • Generate the Expression 1
  • PDF version
  • Generate the Expression 2
  • Generate the Expression 3

Generate the Expression Word Problems

  • Algebra Word Problems 1
  • Algebra Word Problems 2
  • Algebra Word Problems 3
  • Algebra Word Problems 3 UK Version

Algebra Word Problems Walkthrough Video

This short video walkthrough shows the problems from our Algebra Word Problems Worksheet 2 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, please check out the video below!

  • Calculate the Expression 1
  • Calculate the Expression 2
  • Calculate the Expression 3

Solve the Equation Worksheets

  • Solve the Equation 1
  • Solve the Equation 2
  • Solve the Equation 3

More Recommended Math Worksheets

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Free Algebra Problem Solver

The Mathway Calculator is a great way to solve algebra problems that you can type into a calculator.

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The Mathway problem solver will answer your problem instantly and also give you a link to view each of the steps needed.

If you choose to 'View the steps' you will be directed to the Mathway website where you will be able to see in more detail each of the steps needed to solve the problem. Please note that Mathway may charge you a small fee for this!

  • 6th Grade Distributive Property Worksheets

The sheets on this page have been designed to factorize and expand a range of simple expressions using the distributive property..

  • Expressions and Equations 6th Grade

The sheets on this page have been specially designed for 6th graders and are a great introduction to expressions and equations.

Factorising Quadratic Equations

Are you stuck on a quadratic equation and don't know what to do?

Are you looking for some worksheets on factorising quadratic equations to print out?

Take a look at our support pages on quadratic equations where you will hopefully find what you are looking for.

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  • Algebra Math Games

If you are looking for a fun printable algebra game to play then try out our algebra game page.

You will find a range of algebra games that make learning algebra fun and non-threatening.

The only equipment you need is a scientific calculator, some dice, and a few counters!

PEMDAS Worksheets

The sheets in this section involve using parentheses and exponents in simple calculations.

There are also lots of worksheets designed to practice and learn about PEMDAS.

Using these worksheets will help your child to:

  • know and understand how parentheses works;
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  • understand and use PEMDAS to solve a range of problems.
  • PEMDAS Problems Worksheets 5th Grade
  • 6th Grade Order of Operations

Interactive Equality Explorer

This interactive equality explorer has been produced by PhET Interactive Simulations at the University of Colorado.

It is a useful tool for exploring different ideas including negative numbers and algebra equations and equality.

Probably the most useful part of the app is to use the 'Solve It' section once you are confident how it works.

You can then select your level of difficulty and start solving some algebraic equations by getting your variables onto one side of the equation and the numerical values on the other, and then multiplying or dividing the equation until you find the value of the required variable.

interactive equality explorer by PhET

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algebra worksheet 2 5 answer key

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Home > Math Worksheets > Algebra Worksheets > Two Step Equations

Two step math equations are algebraic problems that require you to make two moves to find the value of the unknown variable. For example, using the equation 3x + 5 = 11 we will need to perform two steps to find the value of x. The first step would be to get the constant values of the equation by themselves. In this case 5 and 11 are our constants. We can always perform any operation to the equation as long as we perform the same operation to both sides of the equation. If we want to get the 5 and 11 together, we can perform the opposite operation that is already taking place with one of them. If we were to subtract 5 from both sides, we would be left with 3x = 6. That is step one. For step two we need to get rid of the 3 that is next to the variable. To undo multiplication, we can divide. If we were to divide both sides by 3, we would be left with x = 2. When we start out with algebraic equations, most teacher prefer to name the unknown variable x and stay consistent with its use. Equations of lines often are most frequently solved in two steps.

In this section, your students will work on solving for two variables in algebraic expressions and graphing the results. This set of worksheets introduces your students to the concept of solving for two variables, and provides examples, short practice sets, longer sets of questions, and quizzes. In everyday math, in the real world, a survey of daily tasks of over 1,200 was issued by UMASS. They found that two-step math problems were the most common mathematical tasks people were performing. Utilize these worksheets to demonstrate how to solve two step problems. Students will learn how to create equations from number sentences and solve them.

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Print two step equations worksheets, click the buttons to print each worksheet and associated answer key., solving two step equation problems worksheet 1.

Solve for x in the following 12 problems. Example: 2x + 4 = 12

Print Now!

Worksheet 2

Remember to flip the symbol of the constant that has an operation first. Example: 7x + 4 = 46

Print

Two Step Problems Worksheet 1 (contains negatives)

Do not let the negative value throw you off the scent of the answer. Example: - 4x + 4 = - 12

Worksheet 2 (contains negatives)

You can reorder the problem before you start working on it. Example: - 3 + 7x = 4

Worksheet 3 (contains negatives)

Solve for x in the following 12 problems. Example: 3x + 4 = - 8

More Basic Worksheet 4

The arrange of values is the key here. Example: 5 - 3x = 8

Worksheet 5

See how fast you can complete all of these problems. Example: 3x + 4 = 19

Worksheet 6

See if you can start with the variable first. It is a good way to make sure you master this. Example: 3 + 2x = 17

Two Step Problems Worksheet 7

Solve for x in the following 12 problems. Example: 7 + 2x = 19

Worksheet 8

More practice to make sure you know what you are doing. Example: 6x + 1 = 13

Decimals- Meet the Skill

Learn how to solve the problem: x/2 + 3.5 = 21 Check by substituting your solution to the equation.

Try the Skill

Write an equation and solve for this number sentence: "When a number is divided by 5 and the result added to 35, the result is 105."

Practice the Skill

Solve the equations. Check by substituting your solution to the equation. Example: 9 = a/4+ 4

Practice the Skill Twice

Write an equation and solve for the following 10 sentences.

Show the Skill

These turn up the heat and are more difficult. Example: -13 = -5x + 7

Setup Equations Warm Up

Write an equation and solve for the following 3 sentences.

Meet the Skill

Define the variable, write the equation and solve it: "Three more than two times a number is forty-three."

Where is the variable? Create an equation and solve it: "Two less than a number divided by 5 is eight."

For the following 10 problems locate the unknown variable and make an equation. Oh yeah, solve it too! Example: "Seven less than a number divided by 3 is five."

What final value is being described by the math sentences? Example: "Two more than a number divided by 3 is eleven."

This problems are great to help you start thinking algebraically.

Why not read the problem and take an educated guess before you break down the algebra?

Writing Two-Step Equations - Lesson

We show you how to complete all the following problems.

Practice Worksheet 1

You don't always need to use the variable x. Try using something more abstract to make it interesting.

It is always a good idea to write the components of the equation in the same order as the sentence.

Review Sheet

This walks you through all the steps you need to know.

Practice this skill by completing the 6 problems on this sheet.

See how well you know this topic.

"Do Now"

This is designed to be used as a whole class activity.

  • 5.2 Solving Systems of Equations by Substitution
  • Introduction
  • 1.1 Introduction to Whole Numbers
  • 1.2 Use the Language of Algebra
  • 1.3 Add and Subtract Integers
  • 1.4 Multiply and Divide Integers
  • 1.5 Visualize Fractions
  • 1.6 Add and Subtract Fractions
  • 1.7 Decimals
  • 1.8 The Real Numbers
  • 1.9 Properties of Real Numbers
  • 1.10 Systems of Measurement
  • Key Concepts
  • Review Exercises
  • Practice Test
  • 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
  • 2.2 Solve Equations using the Division and Multiplication Properties of Equality
  • 2.3 Solve Equations with Variables and Constants on Both Sides
  • 2.4 Use a General Strategy to Solve Linear Equations
  • 2.5 Solve Equations with Fractions or Decimals
  • 2.6 Solve a Formula for a Specific Variable
  • 2.7 Solve Linear Inequalities
  • 3.1 Use a Problem-Solving Strategy
  • 3.2 Solve Percent Applications
  • 3.3 Solve Mixture Applications
  • 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
  • 3.5 Solve Uniform Motion Applications
  • 3.6 Solve Applications with Linear Inequalities
  • 4.1 Use the Rectangular Coordinate System
  • 4.2 Graph Linear Equations in Two Variables
  • 4.3 Graph with Intercepts
  • 4.4 Understand Slope of a Line
  • 4.5 Use the Slope-Intercept Form of an Equation of a Line
  • 4.6 Find the Equation of a Line
  • 4.7 Graphs of Linear Inequalities
  • 5.1 Solve Systems of Equations by Graphing
  • 5.3 Solve Systems of Equations by Elimination
  • 5.4 Solve Applications with Systems of Equations
  • 5.5 Solve Mixture Applications with Systems of Equations
  • 5.6 Graphing Systems of Linear Inequalities
  • 6.1 Add and Subtract Polynomials
  • 6.2 Use Multiplication Properties of Exponents
  • 6.3 Multiply Polynomials
  • 6.4 Special Products
  • 6.5 Divide Monomials
  • 6.6 Divide Polynomials
  • 6.7 Integer Exponents and Scientific Notation
  • 7.1 Greatest Common Factor and Factor by Grouping
  • 7.2 Factor Trinomials of the Form x2+bx+c
  • 7.3 Factor Trinomials of the Form ax2+bx+c
  • 7.4 Factor Special Products
  • 7.5 General Strategy for Factoring Polynomials
  • 7.6 Quadratic Equations
  • 8.1 Simplify Rational Expressions
  • 8.2 Multiply and Divide Rational Expressions
  • 8.3 Add and Subtract Rational Expressions with a Common Denominator
  • 8.4 Add and Subtract Rational Expressions with Unlike Denominators
  • 8.5 Simplify Complex Rational Expressions
  • 8.6 Solve Rational Equations
  • 8.7 Solve Proportion and Similar Figure Applications
  • 8.8 Solve Uniform Motion and Work Applications
  • 8.9 Use Direct and Inverse Variation
  • 9.1 Simplify and Use Square Roots
  • 9.2 Simplify Square Roots
  • 9.3 Add and Subtract Square Roots
  • 9.4 Multiply Square Roots
  • 9.5 Divide Square Roots
  • 9.6 Solve Equations with Square Roots
  • 9.7 Higher Roots
  • 9.8 Rational Exponents
  • 10.1 Solve Quadratic Equations Using the Square Root Property
  • 10.2 Solve Quadratic Equations by Completing the Square
  • 10.3 Solve Quadratic Equations Using the Quadratic Formula
  • 10.4 Solve Applications Modeled by Quadratic Equations
  • 10.5 Graphing Quadratic Equations in Two Variables

Learning Objectives

By the end of this section, you will be able to:

  • Solve a system of equations by substitution
  • Solve applications of systems of equations by substitution

Be Prepared 5.4

Before you get started, take this readiness quiz.

Simplify −5 ( 3 − x ) −5 ( 3 − x ) . If you missed this problem, review Example 1.136 .

Be Prepared 5.5

Simplify 4 − 2 ( n + 5 ) 4 − 2 ( n + 5 ) . If you missed this problem, review Example 1.123 .

Be Prepared 5.6

Solve for y y : 8 y − 8 = 32 − 2 y 8 y − 8 = 32 − 2 y If you missed this problem, review Example 2.34 .

Be Prepared 5.7

Solve for x x : 3 x − 9 y = −3 3 x − 9 y = −3 If you missed this problem, review Example 2.65 .

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.

In this section, we will solve systems of linear equations by the substitution method.

Solve a System of Equations by Substitution

We will use the same system we used first for graphing.

We will first solve one of the equations for either x or y . We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.

Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!

After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.

We’ll fill in all these steps now in Example 5.13 .

Example 5.13

How to solve a system of equations by substitution.

Solve the system by substitution. { 2 x + y = 7 x − 2 y = 6 { 2 x + y = 7 x − 2 y = 6

Try It 5.25

Solve the system by substitution. { −2 x + y = −11 x + 3 y = 9 { −2 x + y = −11 x + 3 y = 9

Try It 5.26

Solve the system by substitution. { x + 3 y = 10 4 x + y = 18 { x + 3 y = 10 4 x + y = 18

Solve a system of equations by substitution.

  • Step 1. Solve one of the equations for either variable.
  • Step 2. Substitute the expression from Step 1 into the other equation.
  • Step 3. Solve the resulting equation.
  • Step 4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
  • Step 5. Write the solution as an ordered pair.
  • Step 6. Check that the ordered pair is a solution to both original equations.

If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in Example 5.14 .

Example 5.14

Solve the system by substitution.

{ x + y = −1 y = x + 5 { x + y = −1 y = x + 5

The second equation is already solved for y . We will substitute the expression in place of y in the first equation.

Try It 5.27

Solve the system by substitution. { x + y = 6 y = 3 x − 2 { x + y = 6 y = 3 x − 2

Try It 5.28

Solve the system by substitution. { 2 x − y = 1 y = −3 x − 6 { 2 x − y = 1 y = −3 x − 6

If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y .

Example 5.15

Solve the system by substitution. { 3 x + y = 5 2 x + 4 y = −10 { 3 x + y = 5 2 x + 4 y = −10

We need to solve one equation for one variable. Then we will substitute that expression into the other equation.

Try It 5.29

Solve the system by substitution. { 4 x + y = 2 3 x + 2 y = −1 { 4 x + y = 2 3 x + 2 y = −1

Try It 5.30

Solve the system by substitution. { − x + y = 4 4 x − y = 2 { − x + y = 4 4 x − y = 2

In Example 5.15 it was easiest to solve for y in the first equation because it had a coefficient of 1. In Example 5.16 it will be easier to solve for x .

Example 5.16

Solve the system by substitution. { x − 2 y = −2 3 x + 2 y = 34 { x − 2 y = −2 3 x + 2 y = 34

We will solve the first equation for x x and then substitute the expression into the second equation.

Try It 5.31

Solve the system by substitution. { x − 5 y = 13 4 x − 3 y = 1 { x − 5 y = 13 4 x − 3 y = 1

Try It 5.32

Solve the system by substitution. { x − 6 y = −6 2 x − 4 y = 4 { x − 6 y = −6 2 x − 4 y = 4

When both equations are already solved for the same variable, it is easy to substitute!

Example 5.17

Solve the system by substitution. { y = −2 x + 5 y = 1 2 x { y = −2 x + 5 y = 1 2 x

Since both equations are solved for y , we can substitute one into the other.

Try It 5.33

Solve the system by substitution. { y = 3 x − 16 y = 1 3 x { y = 3 x − 16 y = 1 3 x

Try It 5.34

Solve the system by substitution. { y = − x + 10 y = 1 4 x { y = − x + 10 y = 1 4 x

Be very careful with the signs in the next example.

Example 5.18

Solve the system by substitution. { 4 x + 2 y = 4 6 x − y = 8 { 4 x + 2 y = 4 6 x − y = 8

We need to solve one equation for one variable. We will solve the first equation for y .

Try It 5.35

Solve the system by substitution. { x − 4 y = −4 −3 x + 4 y = 0 { x − 4 y = −4 −3 x + 4 y = 0

Try It 5.36

Solve the system by substitution. { 4 x − y = 0 2 x − 3 y = 5 { 4 x − y = 0 2 x − 3 y = 5

In Example 5.19 , it will take a little more work to solve one equation for x or y .

Example 5.19

Solve the system by substitution. { 4 x − 3 y = 6 15 y − 20 x = −30 { 4 x − 3 y = 6 15 y − 20 x = −30

We need to solve one equation for one variable. We will solve the first equation for x .

Since 0 = 0 is a true statement, the system is consistent. The equations are dependent. The graphs of these two equations would give the same line. The system has infinitely many solutions.

Try It 5.37

Solve the system by substitution. { 2 x − 3 y = 12 −12 y + 8 x = 48 { 2 x − 3 y = 12 −12 y + 8 x = 48

Try It 5.38

Solve the system by substitution. { 5 x + 2 y = 12 −4 y − 10 x = −24 { 5 x + 2 y = 12 −4 y − 10 x = −24

Look back at the equations in Example 5.19 . Is there any way to recognize that they are the same line?

Let’s see what happens in the next example.

Example 5.20

Solve the system by substitution. { 5 x − 2 y = −10 y = 5 2 x { 5 x − 2 y = −10 y = 5 2 x

The second equation is already solved for y , so we can substitute for y in the first equation.

Since 0 = −10 is a false statement the equations are inconsistent. The graphs of the two equation would be parallel lines. The system has no solutions.

Try It 5.39

Solve the system by substitution. { 3 x + 2 y = 9 y = − 3 2 x + 1 { 3 x + 2 y = 9 y = − 3 2 x + 1

Try It 5.40

Solve the system by substitution. { 5 x − 3 y = 2 y = 5 3 x − 4 { 5 x − 3 y = 2 y = 5 3 x − 4

Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

How to use a problem solving strategy for systems of linear equations.

  • Step 1. Read the problem. Make sure all the words and ideas are understood.
  • Step 2. Identify what we are looking for.
  • Step 3. Name what we are looking for. Choose variables to represent those quantities.
  • Step 4. Translate into a system of equations.
  • Step 5. Solve the system of equations using good algebra techniques.
  • Step 6. Check the answer in the problem and make sure it makes sense.
  • Step 7. Answer the question with a complete sentence.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

Example 5.21

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

Try It 5.41

The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

Try It 5.42

The sum of two number is −6. One number is 10 less than the other. Find the numbers.

In the Example 5.22 , we’ll use the formula for the perimeter of a rectangle, P = 2 L + 2 W .

Example 5.22

The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.

Try It 5.43

The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

Try It 5.44

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

For Example 5.23 we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

Example 5.23

The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

We will draw and label a figure.

Try It 5.45

The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

Try It 5.46

The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

Example 5.24

Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her $10,000 + $40 for each training session. How many training sessions would make the salary options equal?

Try It 5.47

Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?

Try It 5.48

Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

Access these online resources for additional instruction and practice with solving systems of equations by substitution.

  • Instructional Video-Solve Linear Systems by Substitution
  • Instructional Video-Solve by Substitution

Practice Makes Perfect

In the following exercises, solve the systems of equations by substitution.

{ 2 x + y = −4 3 x − 2 y = −6 { 2 x + y = −4 3 x − 2 y = −6

{ 2 x + y = −2 3 x − y = 7 { 2 x + y = −2 3 x − y = 7

{ x − 2 y = −5 2 x − 3 y = −4 { x − 2 y = −5 2 x − 3 y = −4

{ x − 3 y = −9 2 x + 5 y = 4 { x − 3 y = −9 2 x + 5 y = 4

{ 5 x − 2 y = −6 y = 3 x + 3 { 5 x − 2 y = −6 y = 3 x + 3

{ −2 x + 2 y = 6 y = −3 x + 1 { −2 x + 2 y = 6 y = −3 x + 1

{ 2 x + 3 y = 3 y = − x + 3 { 2 x + 3 y = 3 y = − x + 3

{ 2 x + 5 y = −14 y = −2 x + 2 { 2 x + 5 y = −14 y = −2 x + 2

{ 2 x + 5 y = 1 y = 1 3 x − 2 { 2 x + 5 y = 1 y = 1 3 x − 2

{ 3 x + 4 y = 1 y = − 2 5 x + 2 { 3 x + 4 y = 1 y = − 2 5 x + 2

{ 3 x − 2 y = 6 y = 2 3 x + 2 { 3 x − 2 y = 6 y = 2 3 x + 2

{ −3 x − 5 y = 3 y = 1 2 x − 5 { −3 x − 5 y = 3 y = 1 2 x − 5

{ 2 x + y = 10 − x + y = −5 { 2 x + y = 10 − x + y = −5

{ −2 x + y = 10 − x + 2 y = 16 { −2 x + y = 10 − x + 2 y = 16

{ 3 x + y = 1 −4 x + y = 15 { 3 x + y = 1 −4 x + y = 15

{ x + y = 0 2 x + 3 y = −4 { x + y = 0 2 x + 3 y = −4

{ x + 3 y = 1 3 x + 5 y = −5 { x + 3 y = 1 3 x + 5 y = −5

{ x + 2 y = −1 2 x + 3 y = 1 { x + 2 y = −1 2 x + 3 y = 1

{ 2 x + y = 5 x − 2 y = −15 { 2 x + y = 5 x − 2 y = −15

{ 4 x + y = 10 x − 2 y = −20 { 4 x + y = 10 x − 2 y = −20

{ y = −2 x − 1 y = − 1 3 x + 4 { y = −2 x − 1 y = − 1 3 x + 4

{ y = x − 6 y = − 3 2 x + 4 { y = x − 6 y = − 3 2 x + 4

{ y = 2 x − 8 y = 3 5 x + 6 { y = 2 x − 8 y = 3 5 x + 6

{ y = − x − 1 y = x + 7 { y = − x − 1 y = x + 7

{ 4 x + 2 y = 8 8 x − y = 1 { 4 x + 2 y = 8 8 x − y = 1

{ − x − 12 y = −1 2 x − 8 y = −6 { − x − 12 y = −1 2 x − 8 y = −6

{ 15 x + 2 y = 6 −5 x + 2 y = −4 { 15 x + 2 y = 6 −5 x + 2 y = −4

{ 2 x − 15 y = 7 12 x + 2 y = −4 { 2 x − 15 y = 7 12 x + 2 y = −4

{ y = 3 x 6 x − 2 y = 0 { y = 3 x 6 x − 2 y = 0

{ x = 2 y 4 x − 8 y = 0 { x = 2 y 4 x − 8 y = 0

{ 2 x + 16 y = 8 − x − 8 y = −4 { 2 x + 16 y = 8 − x − 8 y = −4

{ 15 x + 4 y = 6 −30 x − 8 y = −12 { 15 x + 4 y = 6 −30 x − 8 y = −12

{ y = −4 x 4 x + y = 1 { y = −4 x 4 x + y = 1

{ y = − 1 4 x x + 4 y = 8 { y = − 1 4 x x + 4 y = 8

{ y = 7 8 x + 4 −7 x + 8 y = 6 { y = 7 8 x + 4 −7 x + 8 y = 6

{ y = − 2 3 x + 5 2 x + 3 y = 11 { y = − 2 3 x + 5 2 x + 3 y = 11

In the following exercises, translate to a system of equations and solve.

The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

The sum of two numbers is 30. One number is 4 less than the other. Find the numbers.

The sum of two numbers is −26. One number is 12 less than the other. Find the numbers.

The perimeter of a rectangle is 50. The length is 5 more than the width. Find the length and width.

The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width.

The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.

The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles.

The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.

The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.

Maxim has been offered positions by two car dealers. The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. The second pays a salary of $20,000 plus a commission of $500 for each car sold. How many cars would need to be sold to make the total pay the same?

Jackie has been offered positions by two cable companies. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?

Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

Everyday Math

When Gloria spent 15 minutes on the elliptical trainer and then did circuit training for 30 minutes, her fitness app says she burned 435 calories. When she spent 30 minutes on the elliptical trainer and 40 minutes circuit training she burned 690 calories. Solve the system { 15 e + 30 c = 435 30 e + 40 c = 690 { 15 e + 30 c = 435 30 e + 40 c = 690 for e e , the number of calories she burns for each minute on the elliptical trainer, and c c , the number of calories she burns for each minute of circuit training.

Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system { 56 s = 70 t s = t + 1 2 { 56 s = 70 t s = t + 1 2 .

  • ⓐ for t t to find out how long it will take Tina to catch up to Stephanie.
  • ⓑ what is the value of s s , the number of hours Stephanie will have driven before Tina catches up to her?

Writing Exercises

Solve the system of equations { x + y = 10 x − y = 6 { x + y = 10 x − y = 6

ⓐ by graphing. ⓑ by substitution. ⓒ Which method do you prefer? Why?

Solve the system of equations { 3 x + y = 12 x = y − 8 { 3 x + y = 12 x = y − 8 by substitution and explain all your steps in words.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

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Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
  • Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
  • Publisher/website: OpenStax
  • Book title: Elementary Algebra 2e
  • Publication date: Apr 22, 2020
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
  • Section URL: https://openstax.org/books/elementary-algebra-2e/pages/5-2-solving-systems-of-equations-by-substitution

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  • Discriminant Worksheet
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  • Radical Equations Worksheet

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Answer Key 2.5

  • [latex]x=\pm 8[/latex]
  • [latex]n=\pm 7[/latex]
  • [latex]b=\pm 1[/latex]
  • [latex]x=\pm 2[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrr} 5&+&8a&=&53 \\ -5&&&&-5 \\ \hline &&\dfrac{8a}{8}&=&\dfrac{48}{8} \\ \\ &&a&=&6 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrl} 5&+&8a&=&-53 \\ -5&&&&-5 \\ \hline &&\dfrac{8a}{8}&=&\dfrac{-58}{8} \\ \\ &&a&=&-\dfrac{58}{8}\text{ or }-7\dfrac{1}{4} \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrl} 9n&+&8&=&46 \\ &-&8&&-8 \\ \hline &&\dfrac{9n}{9}&=&\dfrac{38}{9} \\ \\ &&n&=&\dfrac{38}{9}\text{ or }4\dfrac{2}{9} \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 9n&+&8&=&-46 \\ &-&8&&-8 \\ \hline &&\dfrac{9n}{9}&=&\dfrac{-54}{9} \\ \\ &&n&=&-6 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrr} 3k&+&8&=&2 \\ &-&8&&-8 \\ \hline &&\dfrac{3k}{3}&=&\dfrac{-6}{3} \\ \\ &&k&=&-2 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 3k&+&8&=&-2 \\ &-&8&&-8 \\ \hline &&\dfrac{3k}{3}&=&\dfrac{-10}{3} \\ \\ &&k&=&-\dfrac{10}{3} \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrl} 3&-&x&=&\phantom{-}6 \\ -3&&&&-3 \\ \hline &&(-x&=&\phantom{-}3)(-1) \\ &&x&=&-3 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrl} 3&-&x&=&-6 \\ -3&&&&-3 \\ \hline &&(-x&=&-9)(-1) \\ &&x&=&\phantom{-}9 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrl} \dfrac{-7}{-7}\left| -3-3r \right|&=&\dfrac{-21}{-7} \\ |-3-3r|&=&3 \end{array}[/latex] [latex]\phantom{1}[/latex] [latex]\begin{array}{ll} \begin{array}{rrrrr} -3&-&3r&=&3 \\ +3&&&&+3 \\ \hline &&\dfrac{-3r}{-3}&=&\dfrac{6}{-3} \\ \\ &&r&=&-2 \end{array} & \hspace{0.5in} \begin{array}{rrrrr} -3&-&3r&=&-3 \\ +3&&&&+3 \\ \hline &&\dfrac{-3r}{-3}&=&\dfrac{0}{-3} \\ \\ &&r&=&0 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrrrr} |2+2b|&+&1&=&3 \\ &-&1&&-1 \\ \hline |2+2b|&&&=&2 \\ \end{array}[/latex] [latex]\begin{array}[t]{ll} \\ \begin{array}[t]{rrrrr} 2&+&2b&=&2 \\ -2&&&&-2 \\ \hline &&2b&=&0 \\ &&b&=&0 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 2&+&2b&=&-2 \\ -2&&&&-2 \\ \hline &&\dfrac{2b}{2}&=&\dfrac{-4}{2} \\ \\ &&b&=&-2 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrl} \dfrac{7}{7}|-7x-3|&=&\dfrac{21}{7} \\ |-7x-3|&=&3 \end{array}[/latex] [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} -7x&-&3&=&3 \\ &+&3&&+3 \\ \hline &&\dfrac{-7x}{-7}&=&\dfrac{6}{-7} \\ \\ &&x&=&-\dfrac{6}{7} \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} -7x&-&3&=&-3 \\ &+&3&&+3 \\ \hline &&-7x&=&0 \\ &&x&=&0 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrr} -4&-&3n&=&2 \\ +4&&&&+4 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{6}{-3} \\ \\ &&n&=&-2 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} -4&-&3n&=&-2 \\ +4&&&&+4 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{2}{-3} \\ \\ &&n&=&-\dfrac{2}{3} \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrrrrrr} 8|5p &+&8|&-&5&=&11 \\ &&&+&5&&+5 \\ \hline &&\dfrac{8}{8}|5p &+&8|&=&\dfrac{16}{8} \\ &&|5p &+&8|&=&2 \end{array}[/latex] [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} 5p&+&8&=&2 \\ &-&8&&-8 \\ \hline &&\dfrac{5p}{5}&=&\dfrac{-6}{5} \\ \\ &&p&=&-\dfrac{6}{5} \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 5p&+&8&=&-2 \\ &-&8&&-8 \\ \hline &&\dfrac{5p}{5}&=&\dfrac{-10}{5} \\ \\ &&p&=&-2 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrrrrrl} 3&-&|6n&+&7|&=&-40 \\ -3&&&&&&-3 \\ \hline &&(-|6n&+&7|&=&-43)(-1) \\ &&|6n&+&7|&=&43 \end{array}[/latex] [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} 6n&+&7&=&43 \\ &-&7&&-7 \\ \hline &&\dfrac{6n}{6}&=&\dfrac{36}{6} \\ \\ &&n&=&6 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 6n&+&7&=&-43 \\ &-&7&&-7 \\ \hline &&\dfrac{6n}{6}&=&\dfrac{-50}{6} \\ \\ &&n&=&-\dfrac{25}{3} \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrrrrrr} 5|3&+&7m|&+&1&=&51 \\ &&&-&1&&-1 \\ \hline &&\dfrac{5}{5}|3&+&7m|&=&\dfrac{50}{5} \\ &&|3&+&7m|&=&10 \end{array}[/latex] [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} 3&+&7m&=&10 \\ -3&&&&-3 \\ \hline &&\dfrac{7m}{7}&=&\dfrac{7}{7} \\ \\ &&m&=&1 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 3&+&7m&=&-10 \\ -3&&&&-3 \\ \hline &&\dfrac{7m}{7}&=&\dfrac{-13}{7} \\ \\ &&m&=&-\dfrac{13}{7} \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrrrrrr} 4|r&+&7|&+&3&=&59 \\ &&&-&3&&-3 \\ \hline &&\dfrac{4}{4}|r&+&7|&=&\dfrac{56}{4} \\ &&|r&+&7|&=&14 \end{array}[/latex] [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} r&+&7&=&14 \\ &-&7&&-7 \\ \hline &&r&=&7 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} r&+&7&=&-14 \\ &-&7&&-7 \\ \hline &&r&=&-21 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrrrrrr} -7&+&8|-7x&-&3|&=&73 \\ +7&&&&&&+7 \\ \hline &&\dfrac{8}{8}|-7x&-&3|&=&\dfrac{80}{8} \\ &&|-7x&-&3|&=&10 \end{array}[/latex] [latex]\phantom{1}[/latex] [latex]\begin{array}{ll} \begin{array}{rrrrr} -7x&-&3&=&10 \\ &+&3&&+3 \\ \hline &&\dfrac{-7x}{-7}&=&\dfrac{13}{-7} \\ \\ &&x&=&-\dfrac{13}{7} \end{array} & \hspace{0.5in} \begin{array}{rrrrr} -7x&-&3&=&-10 \\ &+&3&&+3 \\ \hline &&\dfrac{-7x}{-7}&=&\dfrac{-7}{-7} \\ \\ &&x&=&1 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{rrrrrrr} 8|3&-&3n|&-&5&=&91 \\ &&&+&5&&+5 \\ \hline &&\dfrac{8}{8}|3&-&3n|&=&\dfrac{96}{8} \\ &&|3&-&3n|&=&12 \end{array}[/latex] [latex]\begin{array}[t]{ll}\\ \begin{array}[t]{rrrrr} 3&-&3n&=&12 \\ -3&&&&-3 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{9}{-3} \\ \\ &&n&=&-3 \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrr} 3&-&3n&=&-12 \\ -3&&&&-3 \\ \hline &&\dfrac{-3n}{-3}&=&\dfrac{-15}{-3} \\ \\ &&n&=&5 \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 5x&+&3&=&2x&-&1 \\ -2x&-&3&&-2x&-&3 \\ \hline &&\dfrac{3x}{3}&=&\dfrac{-4}{3}&& \\ \\ &&x&=&-\dfrac{4}{3}&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrrrr} 5x&+&3&=&-2x&+&1 \\ +2x&-&3&&+2x&-&3 \\ \hline &&\dfrac{7x}{7}&=&\dfrac{-2}{7}&& \\ \\ &&x&=&-\dfrac{2}{7}&& \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 2&+&3x&=&4&-&2x \\ -2&+&2x&&-2&+&2x \\ \hline &&\dfrac{5x}{5}&=&\dfrac{2}{5}&& \\ \\ &&x&=&\dfrac{2}{5}&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrrrr} 2&+&3x&=&-4&+&2x \\ -2&-&2x&&-2&-&2x \\ \hline &&x&=&-6&& \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 3x&-&4&=&2x&+&3 \\ -2x&+&4&&-2x&+&4 \\ \hline &&x&=&7&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrlrr} 3x&-&4&=&-2x&-&3 \\ +2x&+&4&&+2x&+&4 \\ \hline &&\dfrac{5x}{5}&=&\dfrac{1}{5}&& \\ \\ &&x&=&\dfrac{1}{5}&& \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 2x&-&5&=&3x&+&4 \\ -3x&+&5&&-3x&+&5 \\ \hline &&-x&=&9&& \\ &&x&=&-9&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrlrr} 2x&-&5&=&-3x&-&4 \\ +3x&+&5&&+3x&+&5 \\ \hline &&\dfrac{5x}{5}&=&\dfrac{1}{5}&& \\ \\ &&x&=&\dfrac{1}{5}&& \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 4x&-&2&=&6x&+&3 \\ -6x&+&2&&-6x&+&2 \\ \hline &&\dfrac{-2x}{-2}&=&\dfrac{5}{-2}&& \\ \\ &&x&=&-\dfrac{5}{2}&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrrrr} 4x&-&2&=&-6x&-&3 \\ +6x&+&2&&+6x&+&2 \\ \hline &&\dfrac{10x}{10}&=&\dfrac{-1}{10}&& \\ \\ &&x&=&-\dfrac{1}{10}&& \end{array} \end{array}[/latex]
  • [latex]\phantom{a}[/latex] [latex]\begin{array}[t]{ll} \begin{array}[t]{rrrrrrr} 3x&+&2&=&2x&-&3 \\ -2x&-&2&&-2x&-&2 \\ \hline &&x&=&-5&& \end{array} & \hspace{0.5in} \begin{array}[t]{rrrrlrr} 3x&+&2&=&-2x&+&3 \\ +2x&-&2&&+2x&-&2 \\ \hline &&\dfrac{5x}{5}&=&\dfrac{1}{5}&& \\ \\ &&x&=&\dfrac{1}{5}&& \end{array} \end{array}[/latex]

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  • Chapter 1 Linear Functions
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    Free printable Function worksheets (pdf) with answer keys on the domain/range, evaluating functions, composition of functions ,1 to 1 , and more. ... as well as challenge questions at the sheets end. Plus each one comes with an answer key. Domain and Range (Algebra 1 ... Functions Review Worksheet (Algebra 2) Menu; Table of Content; From ...

  15. Glencoe Algebra 2 answers & resources

    ISBN : 0078656095 ISBN-13 : 9780078656095 collections_bookmark Use the table below to find videos, mobile apps, worksheets and lessons that supplement Glencoe Algebra 2. Glencoe Algebra 2 grade 11 workbook & answers help online. Grade: 11, Title: Glencoe Algebra 2, Publisher: Glencoe, ISBN: 0078656095

  16. Two Step Equations Worksheets

    Solving Two Step Equation Problems Worksheet 1 Solve for x in the following 12 problems. Example: 2x + 4 = 12 Worksheet 2 Remember to flip the symbol of the constant that has an operation first. Example: 7x + 4 = 46 Two Step Problems Worksheet 1 (contains negatives) Do not let the negative value throw you off the scent of the answer.

  17. Answer Key Chapter 2

    Introduction to Systems of Equations and Inequalities; 7.1 Systems of Linear Equations: Two Variables; 7.2 Systems of Linear Equations: Three Variables; 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables; 7.4 Partial Fractions; 7.5 Matrices and Matrix Operations; 7.6 Solving Systems with Gaussian Elimination; 7.7 Solving Systems with Inverses; 7.8 Solving Systems with Cramer's Rule

  18. enVision Algebra 2

    Find step-by-step solutions and answers to enVision Algebra 2 - 9780328931590, as well as thousands of textbooks so you can move forward with confidence. ... Key Features of Functions. Section 1-2: Transformations of Functions. Section 1-3: Piecewise-Defined Functions. Section 1-4: Arithmetic Sequences and Series. Section 1-5: Solving Equations ...

  19. 5.2 Solving Systems of Equations by Substitution

    Solve a system of equations by substitution. Step 1. Solve one of the equations for either variable. Step 2. Substitute the expression from Step 1 into the other equation. Step 3. Solve the resulting equation. Step 4. Substitute the solution in Step 3 into one of the original equations to find the other variable.

  20. Quadratic Equation Worksheets (pdfs)

    Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key. Solve Quadratic Equations by Factoring. Solve Quadratic Equations by Completing the Square. Quadratic Formula Worksheets.

  21. Algebra 2 Common Core

    Get Ready! Section 1-1: Patterns and Expressions Section 1-2: Properties of Real Numbers Section 1-3: Algebraic Expressions Page 25: Mid-Chapter Quiz Section 1-4: Solving Equations Section 1-5: Solving Inequalities Section 1-6: Absolute Value Equations and Inequalities Page 51: Chapter Review Page 53: Chapter Test Page 54:

  22. Answer Key 2.5

    Answer Key 5.2. Answer Key 5.3. Answer Key 5.4. Answer Key 5.5. Answer Key 5.6. Answer Key 5.7. Answer Key 6.1. Answer Key 6.2. Answer Key 6.3. Answer Key 6.4. Answer Key 6.5. ... Intermediate Algebra by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

  23. Big Ideas Math Book Algebra 2 Answer Key

    By this, you can finish your homework problems in time. By practicing the problems from our answer key students can prove their best in all types of exams like practice tests, FAs, Quiz, Chapter tests, and so on. Have a look at the list of the chapters given below and start practicing the problems. Chapter 1 Linear Functions.