Systems of Linear Equations
Using a Graph to Solve
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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.
How to Use Graphs to Solve Linear Systems
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 they
lie on the line.
y
(1 , 2)
x
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.
How to Use Graphs to Solve Linear Systems
Consider the following system:
x – y = –1
y
x + 2y = 5
We must ALWAYS verify that your
coordinates actually satisfy both
equations.
(1 , 2)
x
To do this, we substitute the
coordinate (1 , 2) into both
equations.
x – y = –1
(1) – (2) = –1 
x + 2y = 5
(1) + 2(2) =
1+4=5
Since (1 , 2) makes both equations
true, then (1 , 2) is the solution to the
system of linear equations.
Graphing to Solve a Linear System
Solve the following system by graphing:
3x + 6y = 15
–2x + 3y = –3
Start with 3x + 6y = 15
Subtracting 3x from both sides yields
6y = –3x + 15
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.
Dividing everything by 6 gives us…
y= -
1
2
x+
5
2
Similarly, we can add 2x to both
sides and then divide everything by
3 in the second equation to get
y=
2
3
Now, we must graph these two equations
x- 1
Graphing to Solve a Linear System
Solve the following system by graphing:
y
3x + 6y = 15
–2x + 3y = –3
Using the slope intercept forms of these
equations, we can graph them carefully
on graph paper.
y = - 12 x + 52
y=
2
3
x
(3 , 1)
x- 1
Start at the y – intercept, then use the slope.
Label the
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!
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
Solve both equations for y, so that
each equation looks like
y = mx + b.
Step 2: Graph both equations on the
same coordinate plane
Use the slope and y – intercept for
each equation in step 1. Be sure to
use a ruler and graph paper!
Step 3: Estimate where the graphs
intersect.
This is the solution! LABEL the
solution!
Step 4: Check to make sure your
solution makes both equations true.
Substitute the x and y values into both
equations to verify the point is a
solution to both equations.
Graphing to Solve a Linear System
Let's do ONE more…Solve the following system of equations by graphing.
2x + 2y = 3
x – 4y = –1
Step 1: Put both equations in slope –
intercept form
y = - x + 32
y=
1
4
x+
y
LABEL the solution!
(1, 12 )
1
4
x
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.
2(1)+ 2(12 ) = 2 + 1 = 3
1- 4(12 ) = 1- 2 = - 1