Solving Systems of Equations by Substitution
Solving a system by graphing is certainly limited by the accuracy of the graph. If the
lines intersect at a point whose coordinates are not integers, then it is difficult to
determine those coordinates from the graph. The method of solving a system by
substitution does not depend on a graph and is totally accurate. For substitution we
replace a variable in one equation with an equivalent expression obtained from the
other equation. Our intention in this substitution step is to eliminate a variable and
to give us an equation involving only one variable.
Example 1
An independent system solved by substitution
Solve the system by substitution:
2x + 3y = 8
y + 2x = 6
Solution
We can easily solve y + 2x = 6 for y to get y = 2x + 6. Now replace y in the
first equation by 2x + 6:
2x + 3y 
= 8 

2x + 3(2x + 6) 
= 8 
Substitute 2x + 6 for y. 
2x  6x + 18 
= 8 

4x 
= 10 

x 


To find y, we let
in the equation y = 2x + 6:
The next step is to check
and y
= 1 in each equation. If
and y
= 1
in 2x + 3y = 8, we get
If
and y
= 1 in y + 2x = 6, we get
Because both of these equations are true, the solution set to the system is
. The equations of this system are independent.
Example 2
An inconsistent system solved by substitution
Solve by substitution:
x  2y = 3
2x  4y = 7
Solution
Solve the first equation for x to get x  2y = 3 . Substitute 2y + 3 for x in the
second equation:
2x  4y 
= 7 
2(2y + 3) 
= 7 
4y + 6  4y 
= 7 
6 
= 7 
Because 6 = 7 is incorrect no matter what values are chosen for x and y, there is no
solution to this system of equations. The equations are inconsistent. To check, we
write each equation in slopeintercept form:
x  2y 
= 3 
2x  4y 
= 7 
2y 
= x + 3 
4y 
= 2x + 7 
y 

y 

The graphs of these equations are parallel lines with different yintercepts. The
solution set to the system is the empty set, Ã˜.
