Solving Systems of Equations by Graphing

MCC9-12.A.REI.5

Introduction

• The solution to a system of equations is the point or points that make both equations true.

• These systems can have one solution, no solutions, or infinitely many solutions.

• On a graph, the solution to the system is the point of intersection.

Key Concepts

• There are 2 ways that we have looked at graphing linear equations: Create a table or use slope-intercept form y = mx + b.

• Equations not written in slope-intercept form may need to be rewritten. This can be done by solving the equation for y.

Analyzing the Graphs

• If there is one solution to the system of equations, it will be at the point of intersection.

• We call these systems consistent because they have at least one solution. They are also called independent, meaning there is exactly one solution.

Analyzing the Graphs

• Graphs of systems with no solutions have parallel lines. There is no point of intersection. These graphs are referred to as inconsistent.

• Sometimes when you graph the two equations, they actually represent the same line. These systems are referred to as dependent and also consistent because they have at least one solution.

Example 1

• Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If a solution exists, name it.

3

1264

xy

yx

First, solve each equation for y.

4x – 6y = 12

y = -x + 3

3

1264

xy

yx

3

23

2

xy

xy

Now, graph both equations using the slope and y-intercept

Finally, look at the graph

• At one point do the graphs intersect? ______

• So, the solution is the ordered pair ________.

• How could we check to make sure?

Example 2

• Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If a solution exists, name it.

12

448

xy

yx

First, solve each equation for y.

-8x + 4y = 4

y = 2x + 1

12

448

xy

yx

12

12

xy

xy

Now, graph both equations using the slope and y-intercept

Finally, look at the graph

• At one point do the graphs intersect? ______

• So, how many solutions are there?

• How could we check to make sure?

Choose any point on the graph and substitute it into both equations.

Example 3

• Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If a solution exists, name it.

53

826

xy

yx

First, solve each equation for y.

-6x + 2y = 8

y = 3x - 5

53

826

xy

yx

53

43

xy

xy

Now, graph both equations using the slope and y-intercept

Finally, look at the graph

• At one point do the graphs intersect? ______

• So, is there a solution? _______

• Lines are parallel when they have the same slope. Any time lines are parallel, they will never intersect.

Example 4

• You have decided to buy flowers for your mom for Mother’s Day. Each rose costs $4 and each carnation costs $2. You spend a total of $38 and you purchased one dozen flowers. How many of each flower did you buy?

• Because we have not been given the system of equations, we need to come up with equations that model this situation. First, define two variables.

Then, write two equations that appropriately model the situation.

x = # of Rosesy = # of Carnations

12

3824

yx

yx

Now that we have established our system of equations, we can follow the steps from the first few examples.

Solve each equation for y.

4x + 2y = 38 x + y = 12

12

192

xy

xy

Now, graph both equations using the slope and y-intercept

According to the graph, how many of each flower did you buy?

Example 5 - Classwork

• You’ve recently gotten a job at a cell phone company selling phones. There are two salary options. You can get paid $300 per week plus $10 dollars for every phone you sell. The other option is that you get paid $210 per week and $15 dollars for every phone you sell. Which of these two options should you choose and why?