The document discusses solving systems of linear equations by graphing. It explains that a system of linear equations contains two or more linear equations using the same variables. To solve a linear system graphically, the two equations must be put into slope-intercept form and graphed on the same coordinate plane. The point where the lines intersect satisfies both equations and is the solution to the system. The solution must be verified by substituting the coordinates into the original equations.

1. 3.1 SOLVING Systems of Linear
Equations BY GRAPHING
Today’s objectives:
1. I will check solutions of a linear system.
2. I will graph and solve systems of linear equations
in two variables.

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. 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.

4. 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
(1) – (2) = –1  (1) + 2(2) = Since (1 , 2) makes both equations
1+4=5 true, then (1 , 2) is the solution to the
system of linear equations.

5. 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 - intercept y =- 1
2 x+ 5
2
form.
Similarly, we can add 2x to both
sides and then divide everything by
To put the equations in slope 3 in the second equation to get
intercept form, we must solve both
equations for y.
y = 2 x- 1
3
Now, we must graph these two equations.

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

7. 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 - 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 both
solution makes both equations true. equations to verify the point is a
solution to both equations.

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
LABEL the solution!
Step 1: Put both equations in slope -
intercept form. (1, 1)
2
y =- x + 3
2
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

9. •If the lines cross once, there will be
one solution.
(Consistent & Independent)
•If the lines are parallel, there will be
•no solution.
(Inconsistent)
•If the lines are the same, there will be
infinitely many solutions.
(Consistent & Dependent)