Solve a pair of equations using elimination
key notes:
To solve using elimination, follow these four steps:
- Step 1: Make sure the equations have opposite x terms or opposite y terms.
- Step 2: Add to eliminate one variable and solve for the other.
- Step 3: Plug the result of Step 2 into one of the original equations and solve.
- Step 4: State the solution.
Learn with an example
Solve using elimination.
- 8x + 2y = -16
- 8x − 6y = 16
Use elimination to solve the simultaneous equations:
8x + 2y = –16
8x − 6y = 16
Step 1: Make sure the equations have opposite x terms or opposite y terms.
Currently, neither the x terms (8x and 8x) nor the y terms (2y and –6y) are opposites. Use multiplication to rewrite the equations with either opposite x terms or opposite y terms. One good approach is to multiply the first equation by –1.
–(8x + 2y = –16) → –8x − 2y = 16
8x + –6y = 16 → 8x − 6y = 16
Now the x terms (–8x and 8x) are opposites.
Step 2: Add to eliminate one variable and solve for the other.
Add to eliminate the x terms, and then solve for y.
–8x − 2y = 16
+ ( 8x − 6y = 16 )
0x − 8y = 32 —> Add to eliminate the x terms
–8y = 32 —-> Simplify
y = –4 ——> Divide both sides by –8
Step 3: Plug the result of Step 2 into one of the original equations and solve.
Take the result of Step 2, y = -4, and plug it into one of the original equations, such as 8x + 2y = -16. Then find the value of x.
8x + 2y = -16
8x + 2(-4) = -16 —>Plug in y = -4
8x − 8 = -16 ——>Multiply
8x = -8 —–>Add 8 to both sides
x = -1 —->Divide both sides by 8
Step 4: State the solution.
Since x = -1 and y = -4, the solution is (-1, -4).
Solve using elimination.
- x − 2y = -9
- x − 3y = -16
Use elimination to solve the simultaneous equations:
x − 2y = -9x − 3y = -16
Step 1: Make sure the equations have opposite x terms or opposite y terms.
Currently, neither the x terms (x and x) nor the y terms (-2y and -3y) are opposites. Use multiplication to rewrite the equations with either opposite x terms or opposite y terms. One good approach is to multiply the first equation by -1.
| -(x − 2y = -9) | → | -x + 2y = 9 |
| x + -3y = -16 | → | x − 3y = -16 |
Now the x terms (-x and x) are opposites.
Step 2: Add to eliminate one variable and solve for the other.
Add to eliminate the x terms, and then solve for y.
–x + 2y = 9
+ ( x − 3y = –16 )
x − y = –7 —–>Add to eliminate the x terms
–y = –7 Simplify
y = 7 Multiply both sides by –1
Step 3: Plug the result of Step 2 into one of the original equations and solve.
Take the result of Step 2, y = 7, and plug it into one of the original equations, such as x + -2y = -9. Then find the value of x.
x − 2y = -9
x − 2(7) = -9 —>Plug in y = 7
x − 14 = -9 —>Multiply
x = 5 —>Add 14 to both sides
Step 4: State the solution.
Since x = 5 and y = 7, the solution is (5, 7).
Solve using elimination.
-6x – 5y = -15
6x + 9y = 3
( ______ , ______ )
Use elimination to solve the simultaneous equations:
–6x − 5y = –15
6x + 9y = 3
Step 1: Make sure the equations have opposite x terms or opposite y terms.
The x terms (–6x and 6x) are already opposites.
Step 2: Add to eliminate one variable and solve for the other.
Add to eliminate the x terms, and then solve for y.
–6x − 5y = –15
+( 6x + 9y = 3 )
0x + 4y = –12 Add to eliminate the x terms
4y = –12 Simplify
y = –3 Divide both sides by 4
Step 3: Plug the result of Step 2 into one of the original equations and solve.
Take the result of Step 2, y = –3, and plug it into one of the original equations, such as –6x + –5y = –15. Then find the value of x.
–6x − 5y = –15
–6x − 5(–3) = –15 Plugin y = –3
–6x + 15 = –15 Multiply
–6x = –30 Subtract 15 from both sides
x = 5 Divide both sides by –6
Step 4: State the solution.
Since x = 5 and y = –3, the solution is (5, –3).
Let’s practice:
