This document discusses how to solve systems of linear equations by graphing. There are 4 steps: 1) Put both equations in slope-intercept form; 2) Graph both equations on the same coordinate plane; 3) Estimate where the graphs intersect, which is the solution; 4) Check that the solution makes both equations true. It provides examples of solving systems of two linear equations with two variables by graphing them and finding the point where the lines intersect.

1. 7.1 Solve Linear Systems by
Graphing

2. What is a System of Linear Equations?
A system of linear equations is simply two or more linear equations
using the same variables.
We will only be dealing with systems of two equations using two
variables, x and y.
If the system of linear equations is going to have a solution, then
the solution will be an ordered pair (x , y) where x and y make
both equations true at the same time.
We will be working with the graphs of linear systems and how to
find their solutions graphically.

3. Steps to Solve Linear Equations by
Graphing
There are 4 steps to solving a linear system using a graph.
Step 1: Put both equations in slope – Solve both equations for y, so that
intercept form each equation looks like
y = mx + b.
Step 2: Graph both equations on the Use the slope and y – intercept for
same coordinate plane each equation in step 1. Be sure to
use a ruler and graph paper!
Step 3: Estimate where the graphs This is the solution! LABEL the
intersect. solution!
Step 4: Check to make sure your Substitute the x and y values into
solution makes both equations true. both equations to verify the point is a
solution to both equations.

4. How to Use Graphs to Solve Linear Systems
y
Consider the following system:
x – y = –1
x + 2y = 5
Using the graph to the right, we can
see that any of these ordered pairs
will make the first equation true since x
they lie on the line. (1 , 2)
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… The point where they intersect makes both equations true at the same
time.

5. How to Use Graphs to Solve Linear Systems
y
Consider the following system:
x – y = –1
x + 2y = 5
We must ALWAYS verify that your
coordinates actually satisfy both
equations. x
(1 , 2)
To do this, we substitute the
coordinate (1 , 2) into both
equations.
x – y = –1 x + 2y = 5
Since (1 , 2) makes both equations
(1) – (2) = –1  (1) + 2(2) = true, then (1 , 2) is the solution to
1+4=5 the system of linear equations.

6. Graphing to Solve a Linear System
Solve the following system by graphing:
3x + 6y = 15 Start with 3x + 6y = 15
–2x + 3y = –3 Subtracting 3x from both sides yields
6y = –3x + 15
While there are many different Dividing everything by 6 gives us…
ways to graph these equations,
we will be using the slope – y =- 1
2 x+ 2
5
intercept form.
Similarly, we can add 2x to both
sides and then divide everything
STEP 1: To put the equations in by 3 in the second equation to get
slope intercept form, we must y = 2 x- 1
3
solve both equations for y.
Now, we must graph these two equations

7. Graphing to Solve a Linear System
Solve the following system by graphing: y
3x + 6y = 15
–2x + 3y = –3
STEP 2: Using the slope intercept
forms of these equations, we can
x
graph them carefully on graph paper. (3 , 1)
y =- 1 x+ 5
2 2
y = 2 x- 1
3
Start at the y – intercept, then use the
slope.
Lastly, we need to verify our solution is correct, by substituting (3 , 1).

8. Graphing to Solve a Linear System
Let's do ONE more…Solve the following system of equations by graphing.
2x + 2y = 3 y
x – 4y = –1
Step 1: Put both equations in slope –
intercept form (1, 1)
2
y =- x+ 32
y = 1 x+ 1
4 4 x
Step 2: Graph both equations on the
same coordinate plane
Step 3: Estimate where the graphs
intersect. LABEL the solution!
2( 1) + 2 ( 1 ) = 2 +1 = 3
Step 4: Check to make sure your 2
solution makes both equations true. 1- 4 ( 1 ) = 1- 2 = - 1
2