• 5.1 Solve Systems of Equations by Graphing
  • 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
  • 5.2 Solve Systems of Equations by Substitution
  • 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 Quadratic Trinomials with Leading Coefficient 1
  • 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
  • 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

Learning Objectives

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

  • Determine whether an ordered pair is a solution of a system of equations
  • Solve a system of linear equations by graphing
  • Determine the number of solutions of linear system
  • Solve applications of systems of equations by graphing

Be Prepared 5.1

Before you get started, take this readiness quiz.

  • For the equation y = 2 3 x − 4 y = 2 3 x − 4 ⓐ is ( 6 , 0 ) ( 6 , 0 ) a solution? ⓑ is ( −3 , −2 ) ( −3 , −2 ) a solution? If you missed this problem, review Example 2.1 .
  • Find the slope and y -intercept of the line 3 x − y = 12 3 x − y = 12 . If you missed this problem, review Example 4.42 .
  • Find the x - and y -intercepts of the line 2 x − 3 y = 12 2 x − 3 y = 12 . If you missed this problem, review Example 4.21 .

Determine Whether an Ordered Pair is a Solution of a System of Equations

In Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.

Now we will work with systems of linear equations , two or more linear equations grouped together.

System of Linear Equations

When two or more linear equations are grouped together, they form a system of linear equations.

We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.

An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.

A linear equation in two variables, like 2 x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.

To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs ( x , y ) that make both equations true. These are called the solutions to a system of equations .

Solutions of a System of Equations

Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair ( x , y ).

To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Let’s consider the system below:

Is the ordered pair ( 2 , −1 ) ( 2 , −1 ) a solution?

The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.

Let’s try another ordered pair. Is the ordered pair (3, 2) a solution?

The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

Example 5.1

Determine whether the ordered pair is a solution to the system: { x − y = −1 2 x − y = −5 { x − y = −1 2 x − y = −5

ⓐ ( −2 , −1 ) ( −2 , −1 ) ⓑ ( −4 , −3 ) ( −4 , −3 )

Determine whether the ordered pair is a solution to the system: { 3 x + y = 0 x + 2 y = −5 . { 3 x + y = 0 x + 2 y = −5 .

ⓐ ( 1 , −3 ) ( 1 , −3 ) ⓑ ( 0 , 0 ) ( 0 , 0 )

Determine whether the ordered pair is a solution to the system: { x − 3 y = −8 −3 x − y = 4 . { x − 3 y = −8 −3 x − y = 4 .

ⓐ ( 2 , −2 ) ( 2 , −2 ) ⓑ ( −2 , 2 ) ( −2 , 2 )

Solve a System of Linear Equations by Graphing

In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.

The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.

Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.

Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in Figure 5.2 :

For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we’ll use a different method in each section. After seeing the third method, you’ll decide which method was the most convenient way to solve this system.

Example 5.2

How to solve a system of linear equations by graphing.

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

Solve each system by graphing: { x − 3 y = −3 x + y = 5 . { x − 3 y = −3 x + y = 5 .

Solve each system by graphing: { − x + y = 1 3 x + 2 y = 12 . { − x + y = 1 3 x + 2 y = 12 .

The steps to use to solve a system of linear equations by graphing are shown below.

To solve a system of linear equations by graphing.

  • Step 1. Graph the first equation.
  • Step 2. Graph the second equation on the same rectangular coordinate system.
  • Step 3. Determine whether the lines intersect, are parallel, or are the same line.
  • If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.
  • If the lines are parallel, the system has no solution.
  • If the lines are the same, the system has an infinite number of solutions.

Example 5.3

Solve the system by graphing: { y = 2 x + 1 y = 4 x − 1 . { y = 2 x + 1 y = 4 x − 1 .

Both of the equations in this system are in slope-intercept form, so we will use their slopes and y -intercepts to graph them. { y = 2 x + 1 y = 4 x − 1 { y = 2 x + 1 y = 4 x − 1

Solve each system by graphing: { y = 2 x + 2 y = − x − 4 . { y = 2 x + 2 y = − x − 4 .

Solve each system by graphing: { y = 3 x + 3 y = − x + 7 . { y = 3 x + 3 y = − x + 7 .

Both equations in Example 5.3 were given in slope–intercept form. This made it easy for us to quickly graph the lines. In the next example, we’ll first re-write the equations into slope–intercept form.

Example 5.4

Solve the system by graphing: { 3 x + y = −1 2 x + y = 0 . { 3 x + y = −1 2 x + y = 0 .

We’ll solve both of these equations for y y so that we can easily graph them using their slopes and y -intercepts. { 3 x + y = −1 2 x + y = 0 { 3 x + y = −1 2 x + y = 0

Solve each system by graphing: { − x + y = 1 2 x + y = 10 . { − x + y = 1 2 x + y = 10 .

Solve each system by graphing: { 2 x + y = 6 x + y = 1 . { 2 x + y = 6 x + y = 1 .

Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We’ll do this in Example 5.5 .

Example 5.5

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

We will find the x - and y -intercepts of both equations and use them to graph the lines.

Solve each system by graphing: { x + y = 6 x − y = 2 . { x + y = 6 x − y = 2 .

Try It 5.10

Solve each system by graphing: { x + y = 2 x − y = −8 . { x + y = 2 x − y = −8 .

Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.

Example 5.6

Solve the system by graphing: { y = 6 2 x + 3 y = 12 . { y = 6 2 x + 3 y = 12 .

Try It 5.11

Solve each system by graphing: { y = −1 x + 3 y = 6 . { y = −1 x + 3 y = 6 .

Try It 5.12

Solve each system by graphing: { x = 4 3 x − 2 y = 24 . { x = 4 3 x − 2 y = 24 .

In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we’ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.

Example 5.7

Solve the system by graphing: { y = 1 2 x − 3 x − 2 y = 4 . { y = 1 2 x − 3 x − 2 y = 4 .

Try It 5.13

Solve each system by graphing: { y = − 1 4 x + 2 x + 4 y = − 8 . { y = − 1 4 x + 2 x + 4 y = − 8 .

Try It 5.14

Solve each system by graphing: { y = 3 x − 1 6 x − 2 y = 6 . { y = 3 x − 1 6 x − 2 y = 6 .

Example 5.8

Solve the system by graphing: { y = 2 x − 3 −6 x + 3 y = − 9 . { y = 2 x − 3 −6 x + 3 y = − 9 .

Try It 5.15

Solve each system by graphing: { y = − 3 x − 6 6 x + 2 y = − 12 . { y = − 3 x − 6 6 x + 2 y = − 12 .

Try It 5.16

Solve each system by graphing: { y = 1 2 x − 4 2 x − 4 y = 16 . { y = 1 2 x − 4 2 x − 4 y = 16 .

If you write the second equation in Example 5.8 in slope-intercept form, you may recognize that the equations have the same slope and same y -intercept.

When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y -intercept.

Coincident Lines

Coincident lines have the same slope and same y -intercept.

Determine the Number of Solutions of a Linear System

There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.

We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in Example 5.2 through Example 5.6 all had two intersecting lines. Each system had one solution.

A system with parallel lines, like Example 5.7 , has no solution. What happened in Example 5.8 ? The equations have coincident lines , and so the system had infinitely many solutions.

We’ll organize these results in Figure 5.3 below:

Parallel lines have the same slope but different y -intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in Example 5.7 .

The two lines have the same slope but different y -intercepts. They are parallel lines.

Figure 5.4 shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.

Let’s take one more look at our equations in Example 5.7 that gave us parallel lines.

When both lines were in slope-intercept form we had:

Do you recognize that it is impossible to have a single ordered pair ( x , y ) ( x , y ) that is a solution to both of those equations?

We call a system of equations like this an inconsistent system . It has no solution.

A system of equations that has at least one solution is called a consistent system .

Consistent and Inconsistent Systems

A consistent system of equations is a system of equations with at least one solution.

An inconsistent system of equations is a system of equations with no solution.

We also categorize the equations in a system of equations by calling the equations independent or dependent . If two equations are independent equations , they each have their own set of solutions. Intersecting lines and parallel lines are independent.

If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations , we get coincident lines.

Independent and Dependent Equations

Two equations are independent if they have different solutions.

Two equations are dependent if all the solutions of one equation are also solutions of the other equation.

Let’s sum this up by looking at the graphs of the three types of systems. See Figure 5.5 and Figure 5.6 .

Example 5.9

Without graphing, determine the number of solutions and then classify the system of equations: { y = 3 x − 1 6 x − 2 y = 12 . { y = 3 x − 1 6 x − 2 y = 12 .

A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.

Try It 5.17

Without graphing, determine the number of solutions and then classify the system of equations.

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

Try It 5.18

{ y = 1 3 x − 5 x − 3 y = 6 { y = 1 3 x − 5 x − 3 y = 6

Example 5.10

Without graphing, determine the number of solutions and then classify the system of equations: { 2 x + y = − 3 x − 5 y = 5 . { 2 x + y = − 3 x − 5 y = 5 .

A system of equations whose graphs are intersect has 1 solution and is consistent and independent.

Try It 5.19

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

Try It 5.20

{ x + 4 y = 12 − x + y = 3 { x + 4 y = 12 − x + y = 3

Example 5.11

Without graphing, determine the number of solutions and then classify the system of equations. { 3 x − 2 y = 4 y = 3 2 x − 2 { 3 x − 2 y = 4 y = 3 2 x − 2

A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.

Try It 5.21

{ 4 x − 5 y = 20 y = 4 5 x − 4 { 4 x − 5 y = 20 y = 4 5 x − 4

Try It 5.22

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

Solve Applications of Systems of Equations by Graphing

We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We’ll modify the strategy slightly here to make it appropriate for systems of equations.

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.

Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.

Example 5.12

Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. How many quarts of fruit juice and how many quarts of club soda does Sondra need?

Step 1. Read the problem.

We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need.

  Let f = f = number of quarts of fruit juice.      c = c = number of quarts of club soda

We now have the system. { f + c = 10 f = 4 c { f + c = 10 f = 4 c

The point of intersection (2, 8) is the solution. This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice.

Does this make sense in the problem?

Yes, the number of quarts of fruit juice, 8 is 4 times the number of quarts of club soda, 2.

Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda.

Sondra needs 8 quarts of fruit juice and 2 quarts of soda.

Try It 5.23

Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manny need?

Try It 5.24

Alisha is making an 18 ounce coffee beverage that is made from brewed coffee and milk. The number of ounces of brewed coffee is 5 times greater than the number of ounces of milk. How many ounces of coffee and how many ounces of milk does Alisha need?

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

  • Instructional Video Solving Linear Systems by Graphing
  • Instructional Video Solve by Graphing

Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Equations . In the following exercises, determine if the following points are solutions to the given system of equations.

{ 2 x − 6 y = 0 3 x − 4 y = 5 { 2 x − 6 y = 0 3 x − 4 y = 5

ⓐ ( 3 , 1 ) ( 3 , 1 ) ⓑ ( −3 , 4 ) ( −3 , 4 )

{ 7 x − 4 y = −1 −3 x − 2 y = 1 { 7 x − 4 y = −1 −3 x − 2 y = 1

ⓐ   ⓑ ( 1 , −2 ) ( 1 , −2 )

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

ⓐ ( 4 , −3 ) ( 4 , −3 ) ⓑ ( 2 , 0 ) ( 2 , 0 )

{ −3 x + y = 8 − x + 2 y = −9 { −3 x + y = 8 − x + 2 y = −9

ⓐ ( −5 , −7 ) ( −5 , −7 ) ⓑ ( −5 , 7 ) ( −5 , 7 )

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

ⓐ ( 8 7 , 6 7 ) ( 8 7 , 6 7 ) ⓑ ( 1 , 3 4 ) ( 1 , 3 4 )

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

ⓐ ( 5 7 , 2 7 ) ( 5 7 , 2 7 ) ⓑ ( 5 , 2 ) ( 5 , 2 )

{ x + 5 y = 10 y = 3 5 x + 1 { x + 5 y = 10 y = 3 5 x + 1

ⓐ ( −10 , 4 ) ( −10 , 4 ) ⓑ ( 5 4 , 7 4 ) ( 5 4 , 7 4 )

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

ⓐ ( −6 , 5 ) ( −6 , 5 ) ⓑ ( 5 , 4 3 ) ( 5 , 4 3 )

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

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

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

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

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

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

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

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

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

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

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

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

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

{ x − y = 2 2 x − y = 6 { x − y = 2 2 x − y = 6

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

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

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

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

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

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

{ −2 x + 3 y = 3 x + 3 y = 12 { −2 x + 3 y = 3 x + 3 y = 12

{ 2 x − y = 4 2 x + 3 y = 12 { 2 x − y = 4 2 x + 3 y = 12

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

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

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

{ 2 x − 2 y = 8 y = −3 { 2 x − 2 y = 8 y = −3

{ 2 x − y = −1 x = 1 { 2 x − y = −1 x = 1

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

{ x − 3 y = −6 x = −3 { x − 3 y = −6 x = −3

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

{ 4 x − 3 y = 8 8 x − 6 y = 14 { 4 x − 3 y = 8 8 x − 6 y = 14

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

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

{ 3 x + 5 y = 10 y = − 3 5 x + 1 { 3 x + 5 y = 10 y = − 3 5 x + 1

{ x = −3 y + 4 2 x + 6 y = 8 { x = −3 y + 4 2 x + 6 y = 8

{ 4 x = 3 y + 7 8 x − 6 y = 14 { 4 x = 3 y + 7 8 x − 6 y = 14

{ 2 x + y = 6 −8 x − 4 y = −24 { 2 x + y = 6 −8 x − 4 y = −24

{ 5 x + 2 y = 7 −10 x − 4 y = −14 { 5 x + 2 y = 7 −10 x − 4 y = −14

{ x + 3 y = −6 4 y = − 4 3 x − 8 { x + 3 y = −6 4 y = − 4 3 x − 8

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

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

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

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

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

{ y = 1 3 x + 2 x − 3 y = 9 { y = 1 3 x + 2 x − 3 y = 9

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

{ y = 3 x + 4 9 x − 3 y = 18 { y = 3 x + 4 9 x − 3 y = 18

{ y = 2 3 x + 1 2 x − 3 y = 7 { y = 2 3 x + 1 2 x − 3 y = 7

{ 3 x + 4 y = 12 y = −3 x − 1 { 3 x + 4 y = 12 y = −3 x − 1

{ 4 x + 2 y = 10 4 x − 2 y = −6 { 4 x + 2 y = 10 4 x − 2 y = −6

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

{ y = − 1 2 x + 5 x + 2 y = 10 { y = − 1 2 x + 5 x + 2 y = 10

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

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

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

Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water. How many ounces of strawberry juice and how many ounces of water does she need to make 64 ounces of strawberry infused water?

Jamal is making a snack mix that contains only pretzels and nuts. For every ounce of nuts, he will use 2 ounces of pretzels. How many ounces of pretzels and how many ounces of nuts does he need to make 45 ounces of snack mix?

Enrique is making a party mix that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins. How many ounces of nuts and how many ounces of raisins does he need to make 24 ounces of party mix?

Owen is making lemonade from concentrate. The number of quarts of water he needs is 4 times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Owen need to make 100 quarts of lemonade?

Everyday Math

Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant 350 bulbs, how many tulip bulbs and how many daffodil bulbs should he plant?

A marketing company surveys 1,200 people. They surveyed twice as many females as males. How many males and females did they survey?

Writing Exercises

In a system of linear equations, the two equations have the same slope. Describe the possible solutions to the system.

In a system of linear equations, the two equations have the same intercepts. Describe the possible solutions to the system.

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

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

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How to Solve Systems of Equations by Graphing

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  • Solve Systems of Equations by Graphing

Overview of Solve Systems of Equations by Graphing:

How does graphing help solve a system of equations, solving a system of equations by graphing, graphing equations using slope and y-intercept and solving: case 1, graphing equations using slope and y-intercept and solving: case 2, graphing equations using x- and y-intercepts and solving, it’s your turn now, the gist of what we learned so far, graphing tool.

  • Solving Systems of Equations by Graphing - Quiz

When we graph a linear equation on a coordinate plane, we get a straight line. You might wonder how this would help solve a system of linear equations. Now, look at this system.

When the equations are graphed on a grid, we’ll have:

Graph_1

When we consider each line individually, every point on it is the solution of respective equation. But, we need to look for a solution that satisfies both equations simultaneously.

What do you think would be the solution of the system?

You’re right! The point of intersection of the two lines is the solution of the system. Here, the system has a unique solution and that would be (−1, −1) .

What if the lines don’t intersect?

There are two situations, we must consider.

✯ When the lines are parallel, the system has no solution .

✯ When both equations represent the same line, the system has infinitely many solutions .

In this lesson, we’ll deal with graphing and solving systems of equations that have a unique solution .

Let’s look at the step-by-step process of solving a linear system by graphing.

Step 1: Analyze what form each equation of the system is in.

Step 2: Graph the equations using the slope and y-intercept or using the x- and y-intercepts.

  Case 1: If the equations are in the slope-intercept form, identify the slope and y-intercept and graph them.

  Case 2: If one of the equations is in slope-intercept form, rewrite the other one too in that form and graph them.

  Case 3: If both the equations are in other forms, find the x- and y-intercepts and graph them.

Step 3: The ordered pair of the point where the two lines intersect is the required solution.

Let’s explore solving systems of equations in each of the cases.

Here’s a linear system:

The given equations are already in the form y = mx + b, where m is the slope and b is the y-intercept. So, let’s move on to graphing them.

To graph the first equation, we need to use the slope and y-intercept.

The y-intercept in this case is 2. So, let’s plot the point (0, 2) on the coordinate plane.

Graph_2

Now, the slope is −1. So, let’s move down 1 unit and right 1 unit and plot another point.

Graph_3

When we connect these two points drawing a line, we get the graph of the first equation.

Graph_4

Let’s graph the second equation in the similar manner, too.

The y-intercept is 6. So, we need to plot (0, 6) on the grid.

The slope is 1. So, we must move up 1 unit and right 1 unit and plot the other point.

Drawing a line connecting the points, we have:

Graph_5

As you can see, both the lines intersect at a point. Let's label it on our graph.

Graph_6

Now, the ordered pair that corresponds to the point of intersection of the equations is the required solution.

Thus, the solution of the system of equations is (−2, 4) .

Solve y = −x + 7 and y = 2x 3 − 3 graphically.

Both equations are in the slope-intercept form.

Equation 1: m = −1; b = 7

Equation 2: m = 2 3 ; b = −3

Representing the equations graphically we have:

Graph_7

The point of intersection of the two lines is (6, 1).

The solution is (x, y) = (6, 1) .

Check Your Solution! You can cross-check if your solution is right by substituting the x and y values in the equations. Plugging (6, 1) in equation 1, you get: 1 = −6 + 7   1 = 1 ✔ Plugging (6, 1) in equation 2, you get: 1 = 2 (6) 3 − 3 1 = 12 3 − 3 1 = 4 − 3   1 = 1 ✔

Look at this system:

y = −x 2 + 5 2

The former is in the standard form and the latter is in the slope-intercept form. Let’s rearrange and rewrite equation 1 in the form y = mx + c.

Subtracting x from both sides:

 3y = −x

Dividing both sides by 3:

 y = −x 3

So, equation 1 has a slope of −1 3 and y-intercept 0.

Equation 2 has a slope of −1 2 and y-intercept of 5 2 .

Representing the system graphically, we have:

Graph_8

As you can see, the lines cut each other at (15, −5).

Thus, the solution of this system is (x, y) = (15, −5) .

Note: You can also graph the second equation using the x- and y-intercepts rather than rewriting them in slope-intercept form.

If both the equations are rendered in standard form as this system:

−8x + y = −4

We’ll find the x and y-intercepts and solve this system. When we plug y = 0 in the equations, we’ll get the x-intercepts; similarly, when we set x =0, we’ll get the y-intercepts.

Plugging y = 0 in equation 1, we get:

−8x + 0 = −4

−8x −8 = −4 −8   [Dividing both sides by −8]

⇨ x = 1 2

The x-intercept is ( 1 2 , 0) .

Plugging x = 0 in equation 1, we get:

−8(0) + y = −4

⇨ y = −4

The y-intercept is (0, −4) .

Plugging y = 0 in equation 2, we get:

⇨ x = −3

The x-intercept is (−3, 0) .

Plugging x = 0 in equation 2, we get:

−y −1 = −3 −1   [Dividing both sides by −1]

⇨ y = 3

The y-intercept is (0, 3) .

Let’s plot the x- and y-intercepts and graph both the equations.

Graphing equation 1, we get:

Graph_9

Graphing equation 2, we get:

Graph_10

As you can see, both the lines intersect at (1, 4).

Therefore, the solution is (x, y) = (1, 4) .

Note: You can also graph the system by rewriting the equations in slope-intercept form.

Thus, no matter what form the equations are in, all you need to do is graph them and identify the point of intersection.

Refresh

Use the interactive graph below to find the solution of this system.

y = 2x – 3

y = 3x – 2

Drag the point on the line and drop it on the spot you want to plot it.

Solve 2x + y = −1 and y = −x 3 + 4 graphically and check your solution.

Equation 1:

x-intercept = −1 2 ; y-intercept = −1

Equation 2:

Slope = −1 3 ; y-intercept = 4

Graph_11

The solution is (−3, 5) .

Hide answer

View answer

The solution of a system of equations is the point of intersection of the lines obtained when the equations are graphed.

When the lines don’t intersect and are parallel, the system has no solution.

When both the equations give the same line, the system has infinitely many solutions.

No matter which method you use to graph, the solution of a system is unique.

Sample Worksheets

Scale up your skills with our free printable Solving Systems of Linear Equations worksheets !

Use this graphing tool to solve the linear systems in the quiz section. This comes with a provision to alter the scale. Move the points and lines, graph the equations, and solve the system.

Take a Quiz on Solving Systems of Equations by Graphing Now!

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How to solve systems of equations by Graphing

Step by step tutorial for systems of linear equations (in 2 variables)

Video on Solving by Graphing

The Graph Method

What is the solution of the following system of equations?

$$ y = x + 1 \\ y = 2x $$

Graph both equations

picture of systems of equations

On the right, the graph of the two lines

The solution of the system is the point of intersection : (1, 2)

picture of systems of equations

Practice Problems

Use the graph method to solve the system of equations below

$$ y = 2x +1 \\ y = 4x -1 $$

System linear equations answer

The solution of this system is the point of intersection : (1,3).

System linear equations answer

Solve the following system of linear equations by graphing .

$$ \text{ A) } 2y = 4x + 2 \\ \text{ B) }2y = -x + 7 $$

Rewrite each equation in slope intercept form

$$ \text{ A) } 2y = 4x + 2 \\ \frac{1}{2} 2y = \frac{1}{2}(4x+2) \\ y = 2x +1 $$

$$ \text{ B) } 2y = 8x - 2 \\ \frac{1}{2} 2y = \frac{1}{2}( 8x - 2) \\ y = 4x +1 $$

This system of lines is the same system that we looked at in the last example.

Graph each equation to find the point of intersection --which is the solution. (same as earlier problem)

System linear equations picture

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5.2: Solve Systems of Equations by Substitution

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  • Page ID 15152

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

Before you get started, take this readiness quiz.

  • Simplify −5(3−x). If you missed this problem, review Exercise 1.10.43 .
  • Simplify 4−2(n+5). If you missed this problem, review Exercise 1.10.41 .
  • Solve for y. 8y−8=32−2y If you missed this problem, review Exercise 2.3.22 .
  • Solve for x. 3x−9y=−3 If you missed this problem, review Exercise 2.6.22 .

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.

\(\left\{\begin{array}{l}{2 x+y=7} \\ {x-2 y=6}\end{array}\right.\)

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 Exercise \(\PageIndex{1}\).

Exercise \(\PageIndex{1}\): How to Solve a System of Equations by Substitution

Solve the system by substitution. \(\left\{\begin{array}{l}{2 x+y=7} \\ {x-2 y=6}\end{array}\right.\)

This figure has three columns and six rows. The first row says, “Step 1. Solve one of the equations for either variable.” To the right of this, the middl row reads, “We’ll solve the first equation for y.” The third column shows the two equations: 2x + y = 7 and x – 2y = 6. It shows that 2x + y = 7 becomes y = 7 – 2x.

Exercise \(\PageIndex{2}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{-2 x+y=-11} \\ {x+3 y=9}\end{array}\right.\)

Exercise \(\PageIndex{3}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x+3 y=10} \\ {4 x+y=18}\end{array}\right.\)

SOLVE A SYSTEM OF EQUATIONS BY SUBSTITUTION.

  • Solve one of the equations for either variable.
  • Substitute the expression from Step 1 into the other equation.
  • Solve the resulting equation.
  • Substitute the solution in Step 3 into one of the original equations to find the other variable.
  • Write the solution as an ordered pair.
  • 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 Exercise \(\PageIndex{4}\).

Exercise \(\PageIndex{4}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x+y=-1} \\ {y=x+5}\end{array}\right.\)

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

Exercise \(\PageIndex{5}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x+y=6} \\ {y=3 x-2}\end{array}\right.\)

Exercise \(\PageIndex{6}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{2 x-y=1} \\ {y=-3 x-6}\end{array}\right.\)

(−1,−3)

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 .

Exercise \(\PageIndex{7}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{3 x+y=5} \\ {2 x+4 y=-10}\end{array}\right.\)

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

Exercise \(\PageIndex{8}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{4 x+y=2} \\ {3 x+2 y=-1}\end{array}\right.\)

(1,−2)

Exercise \(\PageIndex{9}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{-x+y=4} \\ {4 x-y=2}\end{array}\right.\)

In Exercise \(\PageIndex{7}\) it was easiest to solve for y in the first equation because it had a coefficient of 1. In Exercise \(\PageIndex{10}\) it will be easier to solve for x .

Exercise \(\PageIndex{10}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x-2 y=-2} \\ {3 x+2 y=34}\end{array}\right.\)

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

Exercise \(\PageIndex{11}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x-5 y=13} \\ {4 x-3 y=1}\end{array}\right.\)

(−2,−3)

Exercise \(\PageIndex{12}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x-6 y=-6} \\ {2 x-4 y=4}\end{array}\right.\)

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

Exercise \(\PageIndex{13}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{y=-2 x+5} \\ {y=\frac{1}{2} x}\end{array}\right.\)

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

Exercise \(\PageIndex{14}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{y=3 x-16} \\ {y=\frac{1}{3} x}\end{array}\right.\)

Exercise \(\PageIndex{15}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{y=-x+10} \\ {y=\frac{1}{4} x}\end{array}\right.\)

Be very careful with the signs in the next example.

Exercise \(\PageIndex{16}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{4 x+2 y=4} \\ {6 x-y=8}\end{array}\right.\)

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

Exercise \(\PageIndex{17}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{x-4 y=-4} \\ {-3 x+4 y=0}\end{array}\right.\)

\((2,\frac{3}{2})\)

Exercise \(\PageIndex{18}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{4 x-y=0} \\ {2 x-3 y=5}\end{array}\right.\)

\((−\frac{1}{2},−2)\)

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

Exercise \(\PageIndex{19}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{4 x-3 y=6} \\ {15 y-20 x=-30}\end{array}\right.\)

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

Exercise \(\PageIndex{20}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{2 x-3 y=12} \\ {-12 y+8 x=48}\end{array}\right.\)

infinitely many solutions

Exercise \(\PageIndex{21}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{5 x+2 y=12} \\ {-4 y-10 x=-24}\end{array}\right.\)

Look back at the equations in Exercise \(\PageIndex{22}\). Is there any way to recognize that they are the same line?

Let’s see what happens in the next example.

Exercise \(\PageIndex{22}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{5 x-2 y=-10} \\ {y=\frac{5}{2} x}\end{array}\right.\)

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

Exercise \(\PageIndex{23}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{3 x+2 y=9} \\ {y=-\frac{3}{2} x+1}\end{array}\right.\)

no solution

Exercise \(\PageIndex{24}\)

Solve the system by substitution. \(\left\{\begin{array}{l}{5 x-3 y=2} \\ {y=\frac{5}{3} x-4}\end{array}\right.\)

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.

  • Read the problem. Make sure all the words and ideas are understood.
  • Identify what we are looking for.
  • Name what we are looking for. Choose variables to represent those quantities.
  • Translate into a system of equations.
  • Solve the system of equations using good algebra techniques.
  • Check the answer in the problem and make sure it makes sense.
  • 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?

Exercise \(\PageIndex{25}\)

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

Exercise \(\PageIndex{26}\)

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

The numbers are 3 and 7.

Exercise \(\PageIndex{27}\)

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

The numbers are 2 and −8.

In the Exercise \(\PageIndex{28}\), we’ll use the formula for the perimeter of a rectangle, P = 2 L + 2 W .

Exercise \(\PageIndex{28}\)

Add exercises text here.

Exercise \(\PageIndex{29}\)

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

The length is 12 and the width is 8.

Exercise \(\PageIndex{30}\)

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.

The length is 23 and the width is 6.

For Exercise \(\PageIndex{31}\) 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.

Exercise \(\PageIndex{31}\)

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.

Exercise \(\PageIndex{32}\)

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.

The measure of the angles are 22 degrees and 68 degrees.

Exercise \(\PageIndex{33}\)

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.

The measure of the angles are 36 degrees and 54 degrees.

Exercise \(\PageIndex{34}\)

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?

Exercise \(\PageIndex{35}\)

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?

There would need to be 160 policies sold to make the total pay the same.

Exercise \(\PageIndex{36}\)

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?

Kenneth would need to sell 1,000 suits.

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

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  • xy=10,\:2x+y=1
  • How do you solve a system of equations by substitution?
  • To solve a system of equations by substitution, solve one of the equations for one of the variables, and substitute this expression into the other equation. Then, solve the resulting equation for the remaining variable and substitute this value back into the original equation to find the value of the other variable.
  • How do you solve a system of equations by graphing?
  • To solve a system of equations by graphing, graph both equations on the same set of axes and find the points at which the graphs intersect. Those points are the solutions.
  • How do you solve a system of equations by elimination?
  • To solve a system of equations by elimination, write the system of equations in standard form: ax + by = c, and multiply one or both of the equations by a constant so that the coefficients of one of the variables are opposite. Then, add or subtract the two equations to eliminate one of the variables. Solve the resulting equation for the remaining variable.
  • What are the solving methods of a system of equations?
  • There are several methods for solving a system of equations, including substitution, elimination, and graphing.
  • What is a system of linear equations?
  • A system of linear equations is a system of equations in which all the equations are linear and in the form ax + by = c, where a, b, and c are constants and x and y are variables.

system-of-equations-calculator

  • High School Math Solutions – Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. In this blog post,... Read More

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Infinite Algebra 1

Test and worksheet generator for algebra 1.

Infinite Algebra 1 covers all typical algebra material, over 90 topics in all, from adding and subtracting positives and negatives to solving rational equations. Suitable for any class with algebra content. Designed for all levels of learners from remedial to advanced.

Included in version 2.70.06 released 1/5/2023:

  • Fixed: Choices could appear incorrectly

Included in version 2.70 released 7/8/2022:

  • New: Add preferences for Metric/Imperial units
  • New: Add preference to “Prefer x“ as the variable letter
  • New topic: Discrete Relations
  • New topic: Continuous Relations
  • New topic: Evaluating and Graphing Functions
  • New topic: Direct and Inverse Variation
  • Improved: UI, security, and stability with updated libraries
  • Fixed: Writing Scientific Notation – Choices could be equivalent
  • Fixed: Operations in Scientific Notation – Choices could be equivalent
  • Fixed: Addition and Subtraction in Scientific Notation – Choices could be equivalent

Included in version 2.62 released 2/8/2022:

  • New: Print questions from Presentation View
  • New Topic: Discrete Exponential Growth and Decay Word Problems
  • New Topic: Scientific Notation Add/Subtract
  • Improved: UI, security, and stability with updated framework and libraries
  • Improved: [Mac] Dark mode
  • Fixed: [Windows] - Detached topic list does not close when application is closed

Included in version 2.61.03 released 9/10/2021:

  • Improved: Minor UI improvements
  • Fixed: [Windows] Unable to change page layout to landscape
  • Fixed: [Windows] Error 'Unable to load platform plugin'

Included in version 2.61 released 8/27/2021:

  • Fixed: Preferences page for Kuta Works
  • Improved: Security and stability with updated networking libraries

Included in version 2.60 released 8/19/2021:

  • Improved: Site licenses check expiration date more frequently

Included in version 2.52 released 6/14/2019:

  • New: Scramble questions by directions
  • New: Scramble all questions in assignment
  • New: Consolidate question sets with identical options
  • New: Regroup question sets by directions
  • New: Kuta Works - Create a two semester course
  • New: Kuta Works - Clone assignments from a previous course into a new one
  • Fixed: Presentation View - Arrow keys could change zoom at start/end of assignment
  • Fixed: Tight Layout - Directions could be cramped for no reason
  • Fixed: Inverse Trig - Angles limited for tan( x )
  • Fixed: Inverse Trig - Errors in UI logic

Included in version 2.50 released 4/12/2019:

  • New: Kuta Works - Option to hide answers and results from students until after due date
  • New: Kuta Works - Option to control how long choices are hidden
  • Improved: Options windows appear at a better initial size
  • Improved: Window controls shown at their native size and spacing
  • Improved: Network proxy configuration window
  • Improved: License activation process
  • Fixed: Presentation View - Answers could be cut off
  • Fixed: Some window controls cut off on Windows 10 with display scaling
  • Fixed: Spelling errors
  • Fixed: Kuta Works - Course list could display more than just your active courses
  • Fixed: Writing Scientific Notation - Possible to have duplicate choices

Included in version 2.42 released 12/11/2018:

  • Fixed: Certain symbols incorrect if saved on Mac and loaded on Windows or vice versa

Included in version 2.41 released 9/26/2018:

  • Fixed: Archived courses in Kuta Works are now hidden
  • Fixed: Generating or regenerating questions could cause a crash

Included in version 2.40 released 8/8/2018:

  • New: Integrated with Kuta Works . Now post assignments online.

Included in version 2.25 released 4/27/2018:

  • Improved: Generate questions more quickly
  • Improved: Measurement arrows reach correct endpoints
  • Improved: Added Evaluating Expressions to topic list in index order
  • Improved: Absolute Value Equations - Better decimal numbers
  • Improved: Absolute Value Inequalities - Better decimal numbers
  • Improved: Compound Inequalities - Better decimal numbers
  • Improved: The Distance Formula - Avoid numbers that are too big
  • Improved: Dividing Rational Numbers - Better decimal numbers
  • Improved: Multi-Step Equations - Better decimal numbers and more predictable special cases
  • Improved: Multi-Step Inequalities - Better decimal numbers
  • Improved: One-Step Equations - Better decimal numbers
  • Improved: One-Step Inequalities - Better decimal numbers
  • Improved: Two-Step Equations - Better decimal numbers
  • Improved: Two-Step Inequalities - Better decimal numbers
  • Improved: Order of Operations - Better decimal numbers
  • Fixed: Trig units for piecewise functions and ordered pairs
  • Fixed: Literal Equations - Multiple negative signs could lead to equivalent answers

Included in version 2.18 released 7/31/2017:

  • Fixed: Open circles on graphs were filled in when displayed as answer in red
  • Fixed: Typo in sample custom question

Included in version 2.17 released 4/27/2017:

  • Improved: [Windows] "Export to clipboard as bitmap" renders image with better quality
  • Fixed: Center and Spread of Data - Fixed typo

Included in version 2.16.20 released 11/16/2016:

  • Fixed: Graphing Quadratic Inequalities only listed in index order

Included in version 2.16 released 9/16/2016:

  • Fixed: Solving Proportions - Answer occasionally rounded incorrectly
  • Fixed: Numbers with many significant digits could round incorrectly
  • Fixed: Customized question containing an error could not be modified
  • Fixed: Customized question containing an error could cause a crash
  • Fixed: Feedback tool detects Windows 10 properly
  • Fixed: Mouse cursor when drag-dropping a question
  • Fixed: Keyboard shortcut for changing order of question sets in assignment

Included in version 2.15 released 7/14/2016:

  • New Topic: Literal Equations
  • Improved: Reorder question sets by dragging and dropping
  • Improved: Move individual questions within a set by dragging and dropping
  • Improved: Presentation View - Commands now on a movable, dockable toolbar
  • Improved: Added option for large toolbar icons
  • Improved: New icons
  • Fixed: Certain options could result in double parenthesis or parenthesis around a negative sign
  • Fixed: Arrow over vector variable name not red when appropriate
  • Fixed: Solving Systems of Equations by Graphing - Answer could be displayed twice

Included in version 2.11 released 5/13/2016:

  • Improved: [Windows] "Export to clipboard as bitmap" pastes into Microsoft Word at appropriate DPI
  • Fixed: Keyboard shortcut for adding space to question set

Included in version 2.10 released 5/11/2016:

  • New: Preference for including "None of the above" as a choice
  • New: One column layout
  • New: Added support for Infinite Precalculus
  • New: Easily add piecewise functions of graphs in custom questions: Example: piecewise([2x-3] if [x<5], [x-1] if [x >= 5])
  • New: Easily add functions with restricted domains to graphs in custom questions Example: function(x/2, x<0)
  • Improved: More efficient layout of choices
  • Improved: [Mac] Added option for legal paper in page setup
  • Fixed: Spelling and grammatical errors
  • Fixed: Graphing Quadratic Functions - Prevent equivalent choices
  • Fixed: Number Sets - Could accidentally create rational number instead of irrational
  • Fixed: Solving Proportions - Question could be an identity
  • Note: Older versions will not be able to open assignments saved with this version

Included in version 2.06 released 8/3/2015:

  • Improved: Additional scenarios for statistics
  • Fixed: Crash when saving if previously used folder no longer exists

Included in version 2.05 released 7/20/2015:

  • New Topic: Applying Statistical Models
  • New Topic: Scatter Plots
  • Fixed: Presentation View - Displayed wrong directions in 3-up mode
  • Fixed: Presentation View - Lines too long after creating new problems
  • Fixed: [Windows] Uncommon error when using the network

Included in version 2.04.40 released 3/31/2015:

  • Fixed: Activation data unrecognized under special circumstances
  • Fixed: [Windows] Help files broken by previous release

Included in version 2.04.20 released 3/23/2015:

  • Improved: [Windows] "Export to clipboard as bitmap" uses a transparent background

Included in version 2.04 released 3/17/2015:

  • New Topic: Visualizing Data
  • New Topic: Center and Spread of Data
  • Improved: Better support for proxies
  • Improved: Support for HiDPI Mac Retina displays
  • Improved: Print options window - Some settings are now persistent
  • Improved: Presentation View - Use arrow keys to navigate between questions
  • Improved: Question set updates free response/multiple-choice when all questions have changed
  • Improved: [Mac] Activation also checks /Library/Preferences/ks-config.txt for serial numbers
  • Fixed: [Windows] Crash when printing with certain printers
  • Fixed: Crash when loading an assignment with over 80 custom questions
  • Fixed: Custom question answer was never choice A
  • Fixed: Paper size set before printing
  • Fixed: Radical Equations - Radio buttons backwards

Included in version 2.03 released 9/19/2014:

  • Fixed: Preference for question layout mode not correctly loaded
  • Fixed: Issues with automatic spacing and highlighting questions when using tight layout
  • Fixed: In Presentation View, graphing problems display choices more compactly
  • Fixed: Printing blank pages on HP LaserJet 1018, 1020, and 1022 among others

Included in version 2.02 released 8/5/2014:

  • New: Software uses system proxy when necessary
  • Improved: Activation window has taskbar entry
  • Improved: Remembers the last directory used when saving and opening files
  • Fixed: [Windows only] Possible crash when loading an assignment with a lot of custom questions
  • Fixed: [Mac only] Application stops responding to hover events
  • Fixed: Activation from before v2.0 sometimes not recognized
  • Fixed: Export to clipboard sometimes cuts off choices B and D
  • Fixed: Help contents could become unusable
  • Fixed: Word wrap when modifying a question
  • Fixed: Memory leak

Included in version 2.01 released 7/28/2014:

  • Fixed: Could crash when opening an assignment from a different program

Included in version 2.00 released 7/23/2014:

  • New: Print to PDF
  • New: Filter for topic list
  • New: Zoom while viewing on screen
  • New: "More like these" button in presentation mode
  • New: Show where punch holes will go on the page
  • New: Print only an answer sheet
  • New: When printing, page range is enabled
  • New: Page elements are outlined when hovered
  • New: Double-click on directions to change them
  • New: When merging assignments, options to put similar topics together or put them end-to-end
  • New: Highlight and go to the questions in a question set (menu command or shift-click)
  • New: Site licenses can be activated per-machine if run by the administrator
  • Ctrl-Click on a question
  • Shift-Click on a question
  • Ctrl-Shift-Click on a question
  • Shift-Click in a empty area of a page
  • Ctrl-Shift-Click in a empty area of a page
  • Shift-Click on a question set
  • Ctrl-Shift-Click on a question set
  • Improved: Assignment files are drastically smaller
  • Improved: Sidebar is dockable on either side, can be floated, and position reset
  • Improved: Auto-spacing doesn't leave dead space at bottom of each page
  • Improved: Preview when editing directions
  • Improved: Keyboard shortcuts work no matter where the focus is
  • Improved: Scale number of questions window more user-friendly
  • Fixed: Minor bugs in four topics

Included in version 1.56 released 8/14/2013:

  • Fixed: Factoring Quadratic Expressions - Minor UI issue
  • Fixed: Properties of Exponents - Could generate an incorrect answer
  • Fixed: Systems of Equations by Elimination - "No solution" occasionally should have been "Infinite solutions"

Included in version 1.55 released 12/11/2012:

  • New: Graphs can be added to custom questions
  • Improved: Graphing and Graph Paper utility more powerful and easier to use
  • Improved: Support for loading files from Infinite Calculus
  • Improved: Faster save/load
  • Fixed: Answer for Factoring Quadratic Expressions sometimes incorrect
  • Fixed: Custom questions with an illegal expression could freeze the program
  • Fixed: Certain families of functions graphed incorrectly

Included in version 1.53 released 9/11/2012:

  • New: Preference for notation for greatest integer function
  • New: Added maximize/restore button to Presentation View
  • Improved: Help files
  • Improved: Scroll bars
  • Improved: User interface
  • Improved: When choices make a question too tall for a page, some choices are removed
  • Improved: Algebraic simplification routines are now more efficient
  • Improved: Better graphs for: Graphing Absolute Value Equations, Graphing Linear Equations, Graphing Systems of Equations, Graphing Exponential Functions, Graphing Quadratic Functions, Graphing Linear Inequalities, Graphing Systems of Linear Inequalities, Finding Slope
  • Improved: Better number lines for: Graphing Inequalities, One-Step Inequalities, Two-Step Inequalities, Multi-Step Inequalities, Compound Inequalities, Absolute Value Inequalities
  • Fixed: Wrapping to full-page and half-page could be too wide
  • Fixed: Able to graph certain families of functions
  • Fixed: Issue involving loading & regenerating an assignment and answers being hidden
  • Fixed: Program could ask about changing directions too much
  • Fixed: Graphs for Compound Inequalities could omit interval
  • Fixed: Graphing Exponential Functions could freeze
  • Fixed: Writing Linear Equations: Answer can't be line given in question

Included in version 1.52 released 5/29/2012:

  • Improved: More professional radical signs
  • Improved: Graphs of discontinuous functions have breaks and open / closed holes
  • Improved: Even less likely to crash when generating questions
  • Fixed: Word wrapping could skip blank lines, not wrap where appropriate
  • Fixed: Wrapping to full page in a custom question was too wide
  • Fixed: Products of powers of e (like 2 e &sup3;) no longer display with a multiplication dot
  • Fixed: Most diagrams appeared incorrectly in print preview
  • Fixed: Random error message on save or load

Included in version 1.51.02 released on 4/9/2012:

  • Fixed: Minimum Windows version was set incorrectly

Included in version 1.51 released 4/5/2012:

  • New: Added greatest integer function / floor function to custom questions    Example: Find [int(2x)] when [x=3/5]
  • New: Added piecewise functions to custom questions    Example: [eval(f,x)] = [piecewise [x] if [x < 0], [xx] if [x >= 0]]
  • Improved: Faster! Optimized rendering of questions to screen
  • Improved: Faster! Optimized graphing routines
  • Improved: Improved graphing capabilities
  • Improved: Diagrams drawn more smoothly on screen
  • Fixed: Factoring Quadratics - "Unfactorable" expressions were sometimes factorable
  • Fixed: Graphs could omit holes
  • Fixed: Graphs of constant functions or those involving e and π could be incorrect
  • Fixed: Minor indentation issue in custom questions
  • Fixed: Horizontal asymptotes could be drawn beyond a graph's area

Included in version 1.50 released 3/15/2012:

  • New: Presentation View window is resizeable
  • New: Presentation View has option to automatically hide the answers when a new question is displayed
  • New: High-level filter to prevent questions from containing an illegal expression
  • Improved: Faster! Optimized the simplification of mathematical expressions
  • Improved: Faster! Improved undo/redo algorithm
  • Improved: Smaller executable size
  • Improved: "Current Question Sets" list easier to use
  • Improved: Options windows resize more smoothly
  • Fixed: Systems of Equations Word Problems could freeze
  • Fixed: Punctuation in some word problems
  • Fixed: Certain expressions in a custom question would cause the software to crash
  • Fixed: Expressions like root × term would not print a multiplication dot
  • Note: Beginning with this version, Windows XP SP3 is the minimum required version of Windows

Included in version 1.45 released 4/12/2011:

  • New: Assignments from this program can be opened and modified by our other programs          (Assignments saved with v1.45 or greater can be opened by other programs v1.45 or greater)
  • New: Student data fields (name, date, period) can be renamed
  • New: Additional student data field available
  • New: Add & Continue is available when modifying an existing question set
  • New: Link to this details page when a software update is available
  • New: Product serial numbers can be placed in config.txt to facilitate enterprise installations
  • Improved: Better backwards- and future-compatibility
  • Fixed: Add & Continue respects change in problem type
  • Fixed: Assignments with quotes in the title don't prevent Save As
  • Fixed: Quotients could be sometimes be simplified incorrectly
  • Fixed: Multi-Step Equations: Program could freeze
  • Fixed: Properties of Exponents, numeric expressions: Program could freeze
  • Fixed: Two-Step Equations no longer allows denominators of 1
  • Fixed: Two-Step Inequalities no longer allows denominators of 1
  • Fixed: Dividing Radical Expressions: Program could freeze
  • Fixed: Powers of i are correctly displayed

Included in version 1.42 released on 6/28/2010:

  • New: Automatically checks for updates
  • New: List topics by index order or suggested order
  • New Topic: The Discriminant

Included in version 1.40 released 2/8/2010:

  • New Preference: 'Spacious' or 'tight' layout of questions on page
  • Improved: Question sets with one question are more intelligently spaced
  • Improved: Topic Sets of Numbers now explains N, W, Z, Q, R, and I
  • Improved: Graphs with small ranges now look better
  • Improved: Options screen now resizeable
  • Fixed: Directions occasionally orphaned at bottom of page
  • Fixed: Correct numbers changed because zeros at the end of a number were deleted (50 --> 5)

Included in version 1.38.20 released 5/22/2009:

  • Changed: Minor change to license agreement so that renewals extend the termination dateinstead of beginning a new term.

Included in version 1.38 released 4/10/2009:

  • New: Possible to change directions for a set without regenerating questions
  • Improved: Changing directions cascades changes to appropriate neighbors
  • Correction: Answers to Properties of Exponents could have wrong sign
  • Correction: Inequalities were sometimes solved incorrectly
  • Fixed: Software sometimes failed to renumber questions in assignment

Included in version 1.37 released 1/6/2009:

  • New Topic: Factoring out the GCF
  • Fixed: Software crashes when starting if closed when side panel is minimized

Included in version 1.35 released 11/13/2008:

  • Improved: Isolate up to four questions on screen at once
  • Improved: Support for computers with multiple processors
  • Improved: User interface: Toolbar + Preferences
  • Fixed: Obscure error involving selecting a question and scaling the assignment

Included in version 1.32 released 10/1/2008:

  • New: Export individual questions as bitmaps
  • Improved: Custom questions wrap to whole/half page
  • Improved: Insert special symbols and math in custom questions
  • Improved: Insert 'blanks' into math text
  • Improved: Line thickness preference also now controls darkness of grid lines in graphs
  • Fixed: Internet calls would sometimes report wrong error

Included in version 1.30 released 7/30/2008:

  • New: Add & Continue button
  • New: Undo / redo
  • New: Alternate wide screen layout
  • Improved: Can manually enter compound inequalities
  • Improved: Can manually enter all Greek letters, including pi and theta
  • Improved: Better at reaching the theoretical number of possible questions
  • Improved: More professional-looking assignments

Included in version 1.20 released 9/2/2007:

  • New: Single-question display for teaching with LCD projectors and other display systems
  • New: Custom questions can contain math-formatted text
  • New: Manually modify questions that have been automatically generated
  • New: Check for and download future updates from within Infinite Algebra 1
  • New Topic: Graphing exponential functions
  • New Topic: Mixture word problems
  • New Topic: Distance-rate-time word problems
  • New Topic: Work word problems
  • New Topic: Systems of equations word problems
  • New Topic: Finding trig. ratios
  • New Topic: Finding angles using trigonometry
  • New Topic: Finding missing sides using trigonometry
  • New option: Adding/subtracting polynomials: maximum degree can be 2
  • New option: Distance formula questions: given a line segment on a graph
  • New option: Graphing quadratic functions & inequalities: given transformational form
  • New option: Custom questions can now have separate directions
  • Correction: Right edge of bounding box of logarithms correctly calculated
  • Correction: Certain answers are now rounded off instead of truncated
  • Correction: Custom directions are now word-wrapped instead of running off of page
  • Correction: Order of operations topic printed extra parentheses under certain conditions
  • Fixed: Program crashed when loading an assignment with a question set with zero questions

Try Infinite Algebra 1 Today!

Discover the power and flexibility of our software firsthand with a free, 14-day trial. Installation is fast and simple. Within minutes, you can have the software installed and create the precise worksheets you need -- even for today's lesson.

Use each trial for up to 14 days. The trial version is identical to the retail version except that you cannot print to electronic formats such as PDF.

System of Equations: Graphing Method Calculator

Instructions: Use this calculator to solve a system of two linear equations using the graphical method. Please type two valid linear equations in the boxes provided below:

assignment solve each system by graphing

More about the graphing method to solve linear systems

Systems of linear equations are very commonly found in different context of Algebra. The most commonly found systems in basic Algebra courses are 2 by 2 systems, which consist of two lines equations and two variables.

Such two-by-two systems often appear when solving word problems, proportion problems and assignment problems with constraint. Naturally, larger systems (with more variables and equations) also are common, here focus only on 2x2 systems, because those we can graph.

Graphing Method

How to use graphing method

The graphing method consists of representing each of the linear equations as a line on a graph. Then, we need to find the intersection points between two lines , using the observation that the intersection point of the line (if it exists) will the solution of the system.

What happens if the intersection does not exist? That would be case if the lines are parallel without being the same line, in which case, there is no intersection. The rule is clear: when there is no intersection between the lines, there is no solution to the system.

There is a third case that can also happen: The lines could be parallel but actually identical (this is, they are the same line). So, how many intersection points do you have? Yes, your guess right: you have infinite intersection points, which means that you have infinite solutions.

Solving Systems of equations by graphing answers

So, the methodology is simple: You start with a linear system, and the first thing you do is to graph the two linear equations .

Then, you look at the graph and assess whether the lines intersect at one point only (which happens if the lines have different slopes, in which case you have a unique solution.

If not, see if they parallel and different, in which case there are no solutions. Otherwise, if the two lines are equal, then we have infinite solutions.

How do you solve a system of equations on a graphing calculator?

All systems have different ways of working. In this case of this graphing calculator, all you have to do is to type two linear equations, even if they are not completely simplified. The calculator first will try to get the lines into slope-intercept and will provide you with a graph and with an approximated estimate of the solution.

Different calculators will provide different outputs, but the great advantage of this calculator is that it will provide all the steps of the process.

How do you write systems of equations from a graph?

Linear functions are univocally connected. This is, one linear equation is associated with one and one line only, and conversely, a line is associated with one linear equation and one linear equation only.

So, in order to write systems of equations from a graph, you need to work with each line separately. Take one line and identify two points on the line. With those two points you can compute the slope of the line .

Then, with the slope of the line and the y-intercept, you can write the equation of the line in slope-intercept form .

Related Calculators

Function Grapher - Graph Calculator - Mathcracker.com

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IMAGES

  1. 20++ Solving Systems Of Equations By Graphing Worksheets

    assignment solve each system by graphing

  2. Solve System by Graphing 1

    assignment solve each system by graphing

  3. Solve Each System By Graphing Worksheets

    assignment solve each system by graphing

  4. 42 solve each system by graphing worksheet

    assignment solve each system by graphing

  5. How to Solve Systems of Equations by Graphing

    assignment solve each system by graphing

  6. Solved Solving Systems of Equations by Graphing Solve each

    assignment solve each system by graphing

VIDEO

  1. Graph system of equations

  2. LESSON 6.1 SYSTEMS BY GRAPHING

  3. Solving Linear Systems by Graphing Part 3

  4. Solving Systems by Graphing Pre Recorded

  5. Lesson for 07 02 Linear Equations and Their Graphs

  6. 3) Linking Linear Concepts

COMMENTS

  1. 5.1: Solve Systems of Equations by Graphing

    Solve each system by graphing: {−x + y = 1 2x + y = 10 { − x + y = 1 2 x + y = 10. Answer. Exercise 5.1.12 5.1. 12. Solve each system by graphing: {2x + y = 6 x + y = 1 { 2 x + y = 6 x + y = 1. Answer. Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts.

  2. 4.1: Solving Linear Systems by Graphing

    Exercise \(\PageIndex{7}\) Solving Linear Systems. Set up a linear system of two equations and two variables and solve it using the graphing method. The sum of two numbers is; The larger number is \(10\) less than five times the smaller. The difference between two numbers is \(12\) and their sum is \(4\).

  3. Solving Systems of Equations by Graphing

    Students learn ...more ...more Need a custom math course? Visit https://www.MathHelp.com.This lesson covers solving systems of equations by graphing. Students learn to solve a system of li...

  4. Systems of equations with graphing (practice)

    Systems of equations > Systems of equations with graphing Google Classroom Estimate the solution to the system of equations. You can use the interactive graph below to find the solution. { y = − x + 2 y = 3 x − 4 Choose 1 answer: x = 1 2, y = 3 2 A x = 1 2, y = 3 2 x = 5 2, y = 1 2 B x = 5 2, y = 1 2 x = 3 2, y = 1 2 C x = 3 2, y = 1 2

  5. 5.1 Solve Systems of Equations by Graphing

    Try It 5.16. Solve each system by graphing: { y = 1 2 x − 4 2 x − 4 y = 16. If you write the second equation in Example 5.8 in slope-intercept form, you may recognize that the equations have the same slope and same y -intercept. When we graphed the second line in the last example, we drew it right over the first line.

  6. Solving Systems of Equations By Graphing

    This algebra video tutorial explains how to solve systems of equations by graphing. The solution is the point of intersection of the two graphs. Systems of...

  7. Systems of equations with graphing (video)

    When solving systems of linear equations, one method is to graph both equations on the same coordinate plane. The intersection of the two lines represents a solution that satisfies both equations. Other, more mathematical, methods may also be used. Created by Sal Khan. Questions Tips & Thanks Want to join the conversation? Sort by: Top Voted

  8. Graphical Solutions to Systems of Equations

    A 2 × 2 system of linear equations consists of two equations with two variables, such as athe one below: { 2 x + y = 5 x − y = 1 } When graphed, a system of linear equations is two lines. To solve a system of linear equations, figure out if the two lines intersect and if so, at what point. One way to solve a system of equations is by graphing.

  9. How to Solve Systems of Equations by Graphing

    Step 1: Analyze what form each equation of the system is in. Step 2: Graph the equations using the slope and y-intercept or using the x- and y-intercepts. Case 1: If the equations are in the slope-intercept form, identify the slope and y-intercept and graph them.

  10. How to solve systems of equations by Graphing

    Practice 1. Use the graph method to solve the system of equations below $$ y = 2x +1 \\ y = 4x -1 $$

  11. Solving Systems with One Solution Using Graphing

    Let's solve the following systems by graphing. Graph and solve the system: { y = - x + 1 y = 1 2 x − 2. Since both of these equations are written in slope intercept form, graph them by plotting the y − intercept point and using the slope to locate additional points on each line. The equation y = - x + 1, graphed in b l u e, has y − ...

  12. 5.2: Solve Systems of Equations by Substitution

    Solve the system by substitution. {− x + y = 4 4x − y = 2. In Exercise 5.2.7 it was easiest to solve for y in the first equation because it had a coefficient of 1. In Exercise 5.2.10 it will be easier to solve for x. Solve the system by substitution. {x − 2y = − 2 3x + 2y = 34. Solve for x.

  13. Solving Systems of Linear Equations: Graphing: Assignment

    Round your answer to the nearest tenth. B. 2.4 weeks. Solve the system of linear equations by graphing. Round the solution to the nearest tenth as needed. y + 2.3 = 0.45x. -2y = 4.2x - 7.8. A. (2.4, -1.2) A teacher wrote the equation 3y + 12 = 6x on the board. For what value of b would the additional equation 2y = 4x + b form a system of linear ...

  14. IXL

    IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! Learn more.

  15. System of Equations Calculator

    How do you solve a system of equations by graphing? To solve a system of equations by graphing, graph both equations on the same set of axes and find the points at which the graphs intersect. Those points are the solutions. How do you solve a system of equations by elimination?

  16. Infinite Algebra 1

    Infinite Algebra 1 covers all typical algebra material, over 90 topics in all, from adding and subtracting positives and negatives to solving rational equations. Suitable for any class with algebra content. Designed for all levels of learners from remedial to advanced. Beginning Algebra. Verbal expressions. Order of operations. Sets of numbers.

  17. Solving Systems of Equations by Graphing

    Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. ... Solving Systems of Equations by Graphing. Save Copy. Log InorSign Up. Equations of Lines. 1. Slopes 4. Points. 7. Sliders for First Equation. 12. Folder for Second Equation ...

  18. System of Equations Graphing Method Calculator

    All systems have different ways of working. In this case of this graphing calculator, all you have to do is to type two linear equations, even if they are not completely simplified. The calculator first will try to get the lines into slope-intercept and will provide you with a graph and with an approximated estimate of the solution.

  19. IXL

    Virginia state standards. Textbooks. Test prep. Awards. Improve your math knowledge with free questions in "Solve a system of equations by graphing" and thousands of other math skills.

  20. Quiz 1: Solving Linear Systems by Graphing Flashcards

    linear inequality. an open sentence of the form Ax + By + C < 0 or Ax + By + C > 0. The point will lie in the shaded region of both graphs if it is... true. The point will lie outside of the shaded region of both graphs if it is... false. In order for an ordered pair to be a solution of a system, it must satisfy...

  21. 4.1 Solving Systems by Graphing

    A2.5.4 Solve systems of linear equations and inequalities in two variables by substitution, graphing, and use matrices with three variables; Section 4.1 Solving Systems Graphically. Need a tutor? Click this link and get your first session free! ... Corrective Assignment. a2_4.1_ca.pdf: File Size:

  22. Graphing A System of Linear Equations

    Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.