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Mathematics LibreTexts

5.1: Solve Systems of Equations by Graphing

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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

Before you get started, take this readiness quiz.

  • For the equation \(y=\frac{2}{3}x−4\) ⓐ is (6,0) a solution? ⓑ is (−3,−2) a solution? If you missed this problem, review Exercise 2.1.1 .
  • Find the slope and y-intercept of the line 3x−y=12. If you missed this problem, review Exercise 4.5.7 .
  • Find the x- and y-intercepts of the line 2x−3y=12. If you missed this problem, review Exercise 4.3.7 .

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

In the section on 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.

Definition: 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.

\[\begin{cases}{2 x+y=7} \\ {x-2 y=6}\end{cases}\]

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 .

Definition: 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:

\[\begin{cases}{3x−y=7} \\ {x−2y=4}\end{cases}\]

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

This figure begins with a sentence, “We substitute x =2 and y = -1 into both equations.” The first equation shows that 3x minus y equals 7. Then 3 times 2 minus negative, in parentheses, equals 7. Then 7 equals 7 is true. The second equation reads x minus 2y equals 4. Then 2 minus 2 times negative one in parentheses equals 4. Then 4 = 4 is true.

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?

This figure begins with the sentence, “We substitute x equals 3 and y equals 2 into both equations.” The first equation reads 3 times x minus 7equals 7. Then, 3 times 3 minus 2 equals 7. Then 7 = 7 is true. The second equation reads x minus 2y equals 4. The n times 2 minus 2 times 2 = 4. Then negative 2 = 4 is false.

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.

Exercise \(\PageIndex{1}\)

Determine whether the ordered pair is a solution to the system: \(\begin{cases}{x−y=−1} \\ {2x−y=−5}\end{cases}\)

  • (−2,−1)
  • (−4,−3)

This figure shows two bracketed equations. The first is x minus y = negative 1. The second is 2 times x minus y equals negative 5. The sentence, “We substitute x = negative 2 and y = 1 into both equations,” follows. The first equation shows the substitution and reveals that negative 1 = negative 1. The second equation shows the substitution and reveals that 5 do not equal -5. Under the first equation is the sentence, “(negative 2, negative 1) does not make both equations true.” Under the second equation is the sentence, “(negative 2, negative 1) is not a solution.”

Exercise \(\PageIndex{2}\)

Determine whether the ordered pair is a solution to the system: \(\begin{cases}{3x+y=0} \\ {x+2y=−5}\end{cases}\)

  • (1,−3)

Exercise \(\PageIndex{3}\)

Determine whether the ordered pair is a solution to the system: \(\begin{cases}{x−3y=−8} \\ {−3x−y=4}\end{cases}\)

  • (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 \(\PageIndex{1}\):

This figure shows three x y-coordinate planes. The first plane shows two lines which intersect at one point. Under the graph it says, “The lines intersect. Intersecting lines have one point in common. There is one solution to this system.” The second x y-coordinate plane shows two parallel lines. Under the graph it says, “The lines are parallel. Parallel lines have no points in common. There is no solution to this system.” The third x y-coordinate plane shows one line. Under the graph it says, “Both equations give the same line. Because we have just one line, there are infinitely many solutions.”

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.

Exercise \(\PageIndex{4}\): How to Solve a System of Linear Equations by Graphing

Solve the system by graphing: \(\begin{cases}{2x+y=7} \\ {x−2y=6}\end{cases}\)

This table has four rows and three columns. The first column acts as the header column. The first row reads, “Step 1. Graph the first equation.” Then it reads, “To graph the first line, write the equation in slope-intercept form.” The equation reads 2x + y = 7 and becomes y = -2x + 7 where m = -2 and b = 7. Then it shows a graph of the equations 2x + y = 7. The equation x – 2y = 6 is also listed.

Exercise \(\PageIndex{5}\)

Solve each system by graphing: \(\begin{cases}{x−3y=−3} \\ {x+y=5}\end{cases}\)

Exercise \(\PageIndex{6}\)

Solve each system by graphing: \(\begin{cases}{−x+y=1} \\ {3x+2y=12}\end{cases}\)

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.

  • Graph the first equation.
  • Graph the second equation on the same rectangular coordinate system.
  • 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.

Exercise \(\PageIndex{7}\)

Solve the system by graphing: \(\begin{cases}{y=2x+1} \\ {y=4x−1}\end{cases}\)

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

Exercise \(\PageIndex{8}\)

Solve the system by graphing: \(\begin{cases}{y=2x+2} \\ {y=-x−4}\end{cases}\)

(−2,−2)

Exercise \(\PageIndex{9}\)

Solve the system by graphing: \(\begin{cases}{y=3x+3} \\ {y=-x+7}\end{cases}\)

Both equations in Exercise \(\PageIndex{7}\) 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.

Exercise \(\PageIndex{10}\)

Solve the system by graphing: \(\begin{cases}{3x+y=−1} \\ {2x+y=0}\end{cases}\)

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

Exercise \(\PageIndex{11}\)

Solve each system by graphing: \(\begin{cases}{−x+y=1} \\ {2x+y=10}\end{cases}\)

Exercise \(\PageIndex{12}\)

Solve each system by graphing: \(\begin{cases}{ 2x+y=6} \\ {x+y=1}\end{cases}\)

(5,−4)

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 Exercise \(\PageIndex{13}\).

Exercise \(\PageIndex{13}\)

Solve the system by graphing: \(\begin{cases}{x+y=2} \\ {x−y=4}\end{cases}\)

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

Exercise \(\PageIndex{14}\)

Solve each system by graphing: \(\begin{cases}{x+y=6} \\ {x−y=2}\end{cases}\)

Exercise \(\PageIndex{15}\)

Solve each system by graphing: \(\begin{cases}{x+y=2} \\ {x−y=-8}\end{cases}\)

(5,−3)

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

Exercise \(\PageIndex{16}\)

Solve the system by graphing: \(\begin{cases}{y=6} \\ {2x+3y=12}\end{cases}\)

Exercise \(\PageIndex{17}\)

Solve each system by graphing: \(\begin{cases}{y=−1} \\ {x+3y=6}\end{cases}\)

(9,−1)

Exercise \(\PageIndex{18}\)

Solve each system by graphing: \(\begin{cases}{x=4} \\ {3x−2y=24}\end{cases}\)

(4,−6)

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.

Exercise \(\PageIndex{19}\)

Solve the system by graphing: \(\begin{cases}{y=\frac{1}{2}x−3} \\ {x−2y=4}\end{cases}\)

Exercise \(\PageIndex{20}\)

Solve each system by graphing: \(\begin{cases}{y=-\frac{1}{4}x+2} \\ {x+4y=-8}\end{cases}\)

no solution

Exercise \(\PageIndex{21}\)

Solve each system by graphing: \(\begin{cases}{y=3x−1} \\ {6x−2y=6}\end{cases}\)

Exercise \(\PageIndex{22}\)

Solve the system by graphing: \(\begin{cases}{y=2x−3} \\ {−6x+3y=−9}\end{cases}\)

Exercise \(\PageIndex{23}\)

Solve each system by graphing: \(\begin{cases}{y=−3x−6} \\ {6x+2y=−12}\end{cases}\)

infinitely many solutions

Exercise \(\PageIndex{24}\)

Solve each system by graphing: \(\begin{cases}{y=\frac{1}{2}x−4} \\ {2x−4y=16}\end{cases}\)

If you write the second equation in Exercise \(\PageIndex{22}\) 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 Exercise \(\PageIndex{4}\) through Exercise \(\PageIndex{16}\) all had two intersecting lines. Each system had one solution.

A system with parallel lines, like Exercise \(\PageIndex{19}\), has no solution. What happened in Exercise \(\PageIndex{22}\)? The equations have coincident lines , and so the system had infinitely many solutions.

We’ll organize these results in Figure \(\PageIndex{2}\) below:

This table has two columns and four rows. The first row labels each column “Graph” and “Number of solutions.” Under “Graph” are “2 intersecting lines,” “Parallel lines,” and “Same line.” Under “Number of solutions” are “1,” “None,” and “Infinitely many.”

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 Exercise \(\PageIndex{19}\).

\(\begin{array} {cc} & \begin{cases}{y=\frac{1}{2}x−3} \\ {x−2y=4}\end{cases}\\ \text{The first line is in slope–intercept form.} &\text { If we solve the second equation for } y, \text { we get } \\ &x-2 y =4 \\ y = \frac{1}{2}x -3& x-2 y =-x+4 \\ &y =\frac{1}{2} x-2 \\ m=\frac{1}{2}, b=-3&m=\frac{1}{2}, b=-2 \end{array}\)

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

Figure \(\PageIndex{3}\) shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.

This table is entitled “Number of Solutions of a Linear System of Equations.” There are four columns. The columns are labeled, “Slopes,” “Intercepts,” “Type of Lines,” “Number of Solutions.” Under “Slopes” are “Different,” “Same,” and “Same.” Under “Intercepts,” the first cell is blank, then the words “Different” and “Same” appear. Under “Types of Lines” are the words, “Intersecting,” “Parallel,” and “Coincident.” Under “Number of Solutions” are “1 point,” “No Solution,” and “Infinitely many solutions.”

Let’s take one more look at our equations in Exercise \(\PageIndex{19}\) that gave us parallel lines.

\[\begin{cases}{y=\frac{1}{2}x−3} \\ {x−2y=4}\end{cases}\)]

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

\[y=\frac{1}{2} x-3 \quad y=\frac{1}{2} x-2\]

Do you recognize that it is impossible to have a single ordered pair (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 \(\PageIndex{4}\) and Figure \(\PageIndex{5}\).

This figure shows three x y coordinate planes in a horizontal row. The first shows two lines intersecting. The second shows two parallel lines. The third shows two coincident lines.

Exercise \(\PageIndex{25}\)

Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{cases}{y=3x−1} \\ {6x−2y=12}\end{cases}\)

\(\begin{array}{lrrl} \text{We will compare the slopes and intercepts} & \begin{cases}{y=3x−1} \\ {6x−2y=12}\end{cases} \\ \text{of the two lines.} \\ \text{The first equation is already in} \\ \text{slope-intercept form.} \\ & {y = 3x - 1}\\ \text{Write the second equation in} \\ \text{slope–intercept form.} \\ & 6x-2y &=&12 \\ & -2y &=& -6x - 12 \\ &\frac{-2y}{-2} &=& \frac{-6x + 12}{-2}\\ &y&=&3x-6\\\\ \text{Find the slope and intercept of each line.} & y = 3x-1 & y=3x-6 \\ &m = 3 & m = 3 \\&b=-1 &b=-6 \\ \text{Since the slopes are the same andy-intercepts} \\ \text{are different, the lines are parallel.}\end{array}\)

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

Exercise \(\PageIndex{26}\)

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

\(\begin{cases}{y=−2x−4} \\ {4x+2y=9}\end{cases}\)

no solution, inconsistent, independent

Exercise \(\PageIndex{27}\)

\(\begin{cases}{y=\frac{1}{3}x−5} \\ {x-3y=6}\end{cases}\)

Exercise \(\PageIndex{28}\)

Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{cases}{2x+y=−3} \\ {x−5y=5}\end{cases}\)

\(\begin{array}{lrrlrl} \text{We will compare the slopes and intercepts} & \begin{cases}{2x+y=-3} \\ {x−5y=5}\end{cases} \\ \text{of the two lines.} \\ \text{Write the second equation in} \\ \text{slope–intercept form.} \\ &2x+y&=&-3 & x−5y&=&5\\ & y &=& -2x -3 & -5y &=&-x+5 \\ &&&&\frac{-5y}{-5} &=& \frac{-x + 5}{-5}\\ &&&&y&=&\frac{1}{5}x-1\\\\ \text{Find the slope and intercept of each line.} & y &=& -2x-3 & y&=&\frac{1}{5}x-1 \\ &m &=& -2 & m &=& \frac{1}{5} \\&b&=&-3 &b&=&-1 \\ \text{Since the slopes are the same andy-intercepts} \\ \text{are different, the lines are parallel.}\end{array}\)

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

Exercise \(\PageIndex{29}\)

\(\begin{cases}{3x+2y=2} \\ {2x+y=1}\end{cases}\)

one solution, consistent, independent

Exercise \(\PageIndex{30}\)

\(\begin{cases}{x+4y=12} \\ {−x+y=3}\end{cases}\)

Exercise \(\PageIndex{31}\)

Without graphing, determine the number of solutions and then classify the system of equations. \(\begin{cases}{3x−2y=4} \\ {y=\frac{3}{2}x−2}\end{cases}\)

\(\begin{array}{lrrlrl} \text{We will compare the slopes and intercepts of the two lines.}& \begin{cases}{3x−2y} &=&{4} \\ {y}&=&{\frac{3}{2}x−2}\end{cases} \\ \text{Write the second equation in} \\ \text{slope–intercept form.} \\ &3x-2y&=&4 \\ & -2y &=& -3x +4 \\ &\frac{-2y}{-2} &=& \frac{-3x + 4}{-2}\\ &y&=&\frac{3}{2}x-2\\\\ \text{Find the slope and intercept of each line.} &y&=&\frac{3}{2}x-2\\ \text{Since the equations are the same, they have the same slope} \\ \text{and samey-intercept and so the lines are coincident.}\end{array}\)

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

Exercise \(\PageIndex{32}\)

\(\begin{cases}{4x−5y=20} \\ {y=\frac{4}{5}x−4}\end{cases}\)

infinitely many solutions, consistent, dependent

Exercise \(\PageIndex{33}\)

\(\begin{cases}{ −2x−4y=8} \\ {y=−\frac{1}{2}x−2}\end{cases}\)

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.

  • 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.

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

Exercise \(\PageIndex{34}\)

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.

Step 2. Identify what we are looking for.

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

Step 3. Name what we are looking for. Choose variables to represent those quantities.

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

Step 4. Translate into a system of equations.

This figure shows sentences converted into equations. The first sentence reads, “The number of quarts of fruit juice and the number of quarts of club soda is 10. “Number of quarts of fruit juice” contains a curly bracket beneath the phrase with an “f” centered under the bracket. The “And” also contains a curly bracket beneath it and has a plus sign centered beneath it. “Number of quarts of club soda” contains a curly bracket with the variable “c” beneath it. And finally, the phrase “is 10” contains a curly bracket. Under this it reads equals 10. The second sentence reads, “The number of quarts of fruit juice is four times the number of quarts of club soda”. This sentence is set up similarly in that each phrase contains a curly bracket underneath. The variable “f” represents “The number of quarts of fruit juice”. An equal sign represents “is” and “4c” represents four times the number of quarts of club soda.”

We now have the system. \(\begin{cases}{ f+c=10} \\ {f=4c}\end{cases}\)

Step 5. Solve the system of equations using good algebra techniques.

This figure shows two equations and their graph. The first equation is f = 4c where b = 4 and b = 0. The second equation is f + c = 10. f = negative c +10 where b = negative 1 and b = 10. The x y coordinate plane shows a graph of these two lines which intersect at (2, 8).

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

Step 6. Check the answer in the problem and make sure it makes sense.

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.

Step 7. Answer the question with a complete sentence.

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

Exercise \(\PageIndex{35}\)

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?

Manny needs 3 quarts juice concentrate and 9 quarts water.

Exercise \(\PageIndex{36}\)

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?

Alisha needs 15 ounces of coffee and 3 ounces of milk.

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

Key Concepts

  • Identify the solution to the system. 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.
  • Check the solution in both equations.

This table has two columns and four rows. The first row labels each column “Graph” and “Number of solutions.” Under “Graph” are “2 intersecting lines,” “Parallel lines,” and “Same line.” Under “Number of solutions” are “1,” “None,” and “Infinitely many.”

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Slope Intercept Form Graphing Activity

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After graphing each set of lines, students will create a piece of art by outlining each line in black marker and coloring the sections in between each intersecting line. This will complete a graph with a stained glass look and add some beautiful color to your classroom walls when displayed.

The 4 different graphing art options include:

  • Graphing Art #1 Graphing from Slope Intercept Form Equations: With this option, your students will identify the slope and y-intercept of 8 different lines and then graph them. After graphing, they will color in their graphing. This option is best for students who are just beginning to graph using slope and y-intercept and struggling with this concept.
  • Graphing Art #2 Writing and Graphing Lines from Slope-Intercept Form: Students will be given the slope and y-intercept and will need to write the equation in slope-intercept form and then graph the lines. There are 10 equations to be graphed . This is a great option for students just practicing writing equations in slope-intercept form.
  • Graphing Art #3 Writing and Graphing Equations from 2 Points: With this option students are given two points from a line and will need to find the slope. They will then need to write the equation in slope-intercept form. One way of doing this is to use point-slope form. With this option, there are 12 lines that need to be graphed. This option is a good challenge for those students who easily grasp the concepts of writing equations.
  • Graphing Art #4 Mixed Review- Writing and Graphing Equations in Slope-Intercept Form: With this option students will practice writing equations in slope-intercept form from a variety of ways (slope and point, two points, standard form, etc.). There are 12 lines to be graphed . This is a good option for an end of unit review or to challenge students in your class.

Check out some of the great reviews of this math activity below:

  • "Great review on all of the different ways to write a linear equation. I used this as a review prior to covering systems."
  • "Used for sub plans for students to review slope-intercept form. I love that it has the students changing the equations and then graphing -- two skills in one! Thanks!"
  • "Love doing this activity every year and covering my walls!"
  • "Awesome resource to do instead of just a boring worksheet"
  • "Fun artistic project. I used this as one of my centers and the kids really enjoyed being able to apply art to math! "
  • "This was great. I was able to use it as extra credit with my high-level students who quickly got this concept and wanted to do more practice. Love that it included vertical lines as well, that one always seems to stump them for a minute."

You may also be interested in some of my other math activities below:

Slope Project

Real Life Math Project: Investigating Jobs, Salaries, and Budgets

Mathematician Trading Card Project

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Solve Systems of Equations by Graphing Sheet and Key (pdf)

21 questions with answers.

Students will practice Solve equation system of equations graphically.

Example Questions

Directions: Solve equation system of equations graphically.

problem1

Challenge Problem

problem5

Other Details

This is a 4 part worksheet:

  • Part I Model Problems
  • Part II Practice
  • Part III Challenge Problems
  • Part IV Answer Key
  • Systems of Linear Equations
  • Interactive System of Linear Equations

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

Surface area of a Cylinder

Sports Betting

Sports Betting

Results and outcomes for Super Bowl 2024 prop bets

LAS VEGAS, NEVADA - FEBRUARY 11: Head coach Andy Reid of the Kansas City Chiefs is doused with a sports drink after defeating the San Francisco 49ers 25-22 in overtime during Super Bowl LVIII at Allegiant Stadium on February 11, 2024 in Las Vegas, Nevada. (Photo by Michael Reaves/Getty Images)

Super Bowl LVIII ended in dramatic fashion in overtime with the Kansas City Chiefs beating the San Francisco 49ers in overtime, but the final score was just one piece of the puzzle if you’re into Super Bowl prop bets. From the time of the national anthem, to the coin toss and the color of the Gatorade bath (which almost didn’t happen this year!), we have the results of all the major Super Bowl prop bets right here.

We had our own Super Bowl party props sheet and the answers to that are below as well as some others that could have been included in your own versions.

Pregame props

Coin flip result: Heads

Reba McEntire’s national anthem goes over 86.5 seconds .

Taylor Swift was not shown on the CBS broadcast during the national anthem.

Advertisement

Reba McEntire does not sing the national anthem a cappella.

Halftime show props

First song: “U don’t have to call”

Last song: “Yeah!”

Usher did not wear sunglasses during his halftime performance.

First guess performer: Alicia Keys

Total songs performed over 8.5 (must include lyrics) during the halftime show.

Commercials

The result of Gronk’s kick in his “Kick of Destiny 2” FanDuel commercial was a miss .

Soccer star Lionel Messi (Michelob Ultra) appears in a commercial before David Beckham (Uber Eats).

Actress Aubrey Plaza (Mountain Dew) appears in a commercial before Jenna Ortega (Doritos).

In the Nerds “Who is Addison Rae coaching” commercial, Rae was coaching other (It was a Nerds gummy).

go-deeper

Super Bowl 2025 odds: Texans, Eagles and Falcons have biggest swings from last preseason

Player props

First touchdown scorer: Christian McCaffrey

Anytime touchdown scorers: McCaffrey, Jauan Jennings , Marquez Valdes-Scantling , Mecole Hardman

Final touchdown scorer: Hardman

Patrick Mahomes

Over 1.5 touchdowns thrown

Over 262.5 yards passing

Over 36.5 attempts

Over 26.5 yards rushing

Over 12.5 yards longest rush

Over 0.5 interceptions

Longest pass goes over 36.5 yards

Brock Purdy

Under 1.5 touchdowns thrown

Over 247.5 yards passing

Over 30.5 pass attempts

Under 12.5 yards rushing

Under 0.5 interceptions

Longest pass goes under 37.5 yards

George Kittle

Under 50.5 yards receiving

Under 3.5 catches

Under 21.5 yards longest catch

Deebo Samuel

Under 58.5 yards receiving

Under 4.5 catches

Brandon Aiyuk

Under 61.5 yards receiving

Under 24.5 yards longest catch

Rashee Rice

Under 67.5 yards receiving

Under 6.5 catches

Travis Kelce

Over 70.5 yards receiving

Over 6.5 catches

Over 20.5 yards longest catch

Isaiah Pacheco

Under 65.5 yards rushing

Over 87.5 rushing + receiving yards

Over 15.5 rushing attempts

Under 15.5 yards longest rush

Christian McCaffrey

Under 91.5 yards rushing

Over 18.5 rushing attempts

Over 130.5 rushing + receiving yards

Under 17.5 yards longest rush

Postgame props

Super Bowl MVP: Patrick Mahomes

Gatorade color on winning coach Andy Reid: Purple

MAKE IT RAIN!!!! pic.twitter.com/0MEqkpqMHf — Kansas City Chiefs (@Chiefs) February 12, 2024

Miscellaneous

Winning squares: First quarter 0-0; halftime 49ers 0, Chiefs 3; third quarter 49ers 0, Chiefs 3; final score 49ers 2, Chiefs 5

The 25-22 final score isn’t an NFL Scorigami, but it is the first time 2-5 (or 5-2) had hit in any quarter in Super Bowl history.

Total field goals made: over 6.5

Shortest touchdown: over 1.5 yards

Last turnover of the game: Fumble (technically it was a muffed punt, but it was in the official box score as a lost fumble).

Who will have a longer pass completion: Patrick Mahomes or Brock Purdy? Mahomes, with a 52-yard completion to Mecole Hardman.

Will any of the following events occur in the game: Safety, Flea Flicker Attempt, Field Goal Doink, Defense TD, Special Teams TD: No

First team to score: 49ers

First team to reach 10 points: 49ers

First team to reach 20 points: 49ers

First team to score a touchdown: 49ers

First team to punt: Chiefs

Longest field goal: Over 47.5 yards ( Jake Moody hits a Super Bowl-record 55-yard field goal, Harrison Butker also made a 57-yard field goal to break Moody’s record)

First penalty flag of the game was a false start .

Iconic Las Vegas landmark seen on the broadcast first during the game: Luxor pyramid/sphinx

Will SpongeBob be shown on the CBS broadcast during the game? Yes

First person shown sitting directly next to Taylor Swift during the game: Blake Lively

Will Taylor Swift be shown in the stadium before the final whistle wearing clothing that bears Travis Kelce’s name, number, image or likeness? Yes

go-deeper

How the Chiefs stack up among NFL dynasties (and a path past the Patriots): Sando’s Pick Six

(Photo of Andy Reid: Michael Reaves / Getty Images)

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