System of Equations
Elimination
Using addition and subtraction
S. Calahan
Addition
Use elimination to solve the system of equations.
3x – 5y = -16
2x + 5y = 31
Since the coefficients of the y terms,
-5 and 5, are additive inverses, you
can eliminate the y terms by adding
the equations.
ADD
3x – 5y = -16
+ 2x + 5y = 31
5x
= 15
We canceled the y terms leaving only
the x terms on the left hand side.
Now solve for x.
5x = 15
5
5
x=3
Now let’s solve for y.
Solve for y using substitution
remember x = 3
3x – 5y = -16
2x + 5y = 31
Choose one of the original
equations. 2x + 5y = 31
and replace the x with 3.
Substitute and solve
2 (3) + 5y = 31
6 + 5y = 31
-6
-6
5y = 25
5
5 so, y = 5
Since x = 3 and y = 5
The solution set for the
equations
3x – 5y = -16
2x + 5y = 31
is (3,5). This set of
equations has only one
solution.
Subtraction
5s + 2t = 6
9s + 2t = 22
Since the coefficients of the t
terms, 2 and 2, are the
same, you can eliminate the
t terms by subtracting the
equations.
Add the opposite
5s + 2t = 6
-(9s + 2t = 22) becomes
5s + 2t = 6
-9s -2t = -22
Now ADD
5s + 2t = 6
-9s - 2t = -22
- 4s
= - 16
Cancel the 2t and the -2t
Solve for s
- 4s = - 16
-4
-4
s=4
Now solve for t
Substitute s = 4 into an original
equation and solve for t
5s + 2t = 6
9s + 2t = 22
 Either equation will work. I
will use
5s + 2t = 6 ,so 5(4) + 2y = 6
5s + 2t = 6 and s = 4
5(4) + 2t = 6
20 + 2t = 6
-20
= -20
2t = -14
2
2 , then t = -7.
Therefore, the solution set for
5s + 2t = 6
9s + 2t = 22
is (4, -7).