Solving Linear Systems of Equations by Graphing
One way to find the solution of a linear system in two variables is to graph
each equation on the same coordinate axes.
If the lines intersect, the point(s) of intersection is the solution of the
system.
Example 1
Graph each equation to find the solution of this system.
Solution
To graph each equation, first write it in slopeintercept form, y = mx + b.
â€¢ Letâ€™s start with the first equation.
Subtract 3x from both sides. 
3x  2y 2y 
= 8 = 3x + 8 
Divide both sides by 2.
The yintercept is (0, 4).
Plot the point (0, 4). 
y 

The slope is
This is the
ratio
To locate a second point, start at (0, 4), move up 3 (the rise) and then
move right 2 (the run).
Plot the new point (2, 1). Finally, draw the line through (0, 4) and (2, 1).
Each point on this line represents a solution of 3x  2y = 8.
â€¢ The second equation, y = x + 6, is given in the form y = mx + b.
The yintercept is (0, 6). Plot the point (0, 6).
The slope is 1, which can be written as
To locate a second point, start at (0, 6), move down 1 (the rise) and
move right 1 (the run).
Plot the new point (1, 5).
Finally, draw the line through (0, 6) and (1, 5).
Every point on this line represents a solution of y = x + 6.
From the graph, it appears that the lines intersect at the point (4, 2).
The point (4, 2) is a solution of each equation.
Therefore, the solution of the system is (4, 2).
Letâ€™s verify that (4, 2) satisfies both equations.

First equation 

Second equation 
Is
Is
Is 
3x
3(4)
12 
 
 
2y
2(2)
4
8 
= 8 = 8 ?
= 8 ?
= 8 ? Yes 
Is
Is 
y
(2)
2 
= =
= 
x + 6 (4) + 6 ?
2 ? Yes 
Since (4, 2) satisfies both equations, it is the solution of the system.
The solution can be written as x = 4 and y = 2, or simply (4, 2).
A system that has at least one solution is called a consistent system.
