3-1: Solving Systems by Graphing

What is a System of Linear Equations?

Definition: A system of linear equations is simply two or more linear equations using the same variables.

3x + 2y = 2 Equation 1 x + 2y = 6 Equation 2

Example:

How to Use Graphs to Solve Linear Systems

x

yConsider the following system:

x – y = –1

x + 2y = 5Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they lie on the line.

We can also see that any of these points will make the second equation true.

However, there is ONE coordinate that makes both true at the same time…

(1 , 2)

The point where they intersect makes both equations true at the same time.

•Two intersecting lines•Two lines on top of each other•Two parallel lines

Three Possible Outcomesp. 154

EXAMPLE 4 Writing and Using a Linear System (p. 155)

EXAMPLE 4 Step 1: Write linear equations in standard form

Equation 1

Equation 2

y = 1 x + 30

y = x2.5

EXAMPLE 4 Step 2: Graph both equations

•Two intersecting lines = one solution

•(20, 50) appears to be the solution

EXAMPLE 4 Step 4: Check your solution

Point of intersection: (20,50 ).

Substitute 20 and 50 in place of x and y in both equations:

y = x + 30

y =2.5x

Equation 1 checks.

Equation 2 checks.

50 = 20 + 30

50 = 2.5(20)

ANSWER

Break even point in 20 rides

The solution is (20, 50).

Graphing to Solve a Linear System

Let's summarize! There are 4 steps to solving a linear system using a graph.

Step 1: Put both equations in slope - intercept form.

Step 2: Graph both equations on the same coordinate plane.

Step 3: Estimate where the graphs intersect.

Step 4: Check to make sure your solution makes both equations true.

Solve both equations for y, so that each equation looks like

y = mx + b.

Use the slope and y - intercept for each equation in step 1. Be sure to use a ruler and graph paper!

This is the solution! LABEL the solution!

Substitute the x and y values into both equations to verify the point is a solution to both equations.

4x – y = 25

-3x - 2y = -16

Practice: Checking the Solution

x

y

Page 412, #11:

(6 , -1)

We must ALWAYS verify that your coordinates actually satisfy both equations.

Since (6 , -1) makes both equations true, then (6 , -1) is the solution to the system of linear equations.

-3x - 2y = -16

-3(6) - 2(-1) =

-18 + 2 = -16

To do this, we substitute the coordinate (6 , -1) into both equations.

4x – y = 25

4(6) – (-1) =

24 + 1 = 25

EXAMPLE 1 Solve a system graphically

Graph the linear system and estimate the solution. Then check the solution algebraically.

4x + y = 8

2x – 3y = 18

Equation 1

Equation 2

SOLUTION

Begin by graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (3, – 4). You can check this algebraically as follows.

EXAMPLE 1 Solve a system graphically

Equation 1 Equation 2

4x + y = 8

4(3) + (– 4) 8=?

=?12 – 4 8

8 = 8

2x – 3y = 18

=?2(3) – 3( – 4) 18

=?6 + 12 18

18 = 18

The solution is (3, – 4).

SOLUTION

GUIDED PRACTICE Page 153 Example 1

Graph the linear system and estimate the solution. Then check the solution algebraically.1. 3x + 2y = – 4

x + 3y = 13x + 2y = – 4x + 3y = 1

Equation 1Equation 2

Begin by graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (–2, 1). You can check this algebraically as follows.

GUIDED PRACTICE Page 153, Example 1

Equation 1 Equation 23x + 2y = –4

=?–6 + 2 –4

x + 3y = 1

–2 + 3 1=?

1 = 1

The solution is (–2, 1).

=?3(–2) + 2(1) –4

–4 = –4

(–2 ) + 3( 1) 1=?

Page 142, #23

While there are many different ways to graph these equations, we will be using the slope - intercept form.

To put the equations in slope intercept form, we must solve both equations for y.

Start with 3x + 4y = -10

Subtracting 3x from both sides yields

4y = –3x -10

Dividing everything by 6 gives us…3 54 2y x=- -

Similarly, we can add 7x to both sides and then divide everything by -1 in the second equation to get

7 10y x=- +

Now, we must graph these two equations.

Solve the following system by graphing:

3x + 4y = -10

-7x - y = -10

Page 142, #23, cont.

Solve the following system by graphing:

3x + 6y = 15

–2x + 3y = –3

Using the slope intercept form of these equations, we can graph them carefully on graph paper.

x

y

Start at the y – intercept: Note that my scale is 2 on this graph. then use the slope.

3 54 2y x=- -

7 10y x=- +

Page 142, #23, cont.

Can you read the solution? It looks close to (2, -4)

Check to make sure.

x

y

LABEL the solution!

Graphing to Solve a Linear System

Step 1: Put both equations in slope - intercept form.

Step 2: Graph both equations on the same coordinate plane.

Step 3: Estimate where the graphs intersect. LABEL the solution!

Step 4: Check to make sure your solution makes both equations true.

Let's do ONE more…Solve the following system of equations by graphing.

2x + 2y = 3

x – 4y = -1

32y x=- +

1 14 4y x= +

( ) ( )122 1 2 2 1 3+ = + =

( )121 4 1 2 1- = - =-

( )121,