Math 1 · Unit 6 · Lesson 3 · part 3 of 5
The repeatable procedure
The y coefficients are 2 and −1.
The repeatable procedure
- 1Line them upx terms over x terms, y terms over y terms, numbers over numbers. Rewrite into Ax + By = C form if needed.
- 2Make opposite coefficients, if neededMultiply one or both equations by a number (both sides, Unit 1) so one variable's coefficients are opposites, like 3 and −3.
- 3Add the equationsThat variable's terms add to 0 and it disappears.
- 4Solve the one-variable equationUnit 1.
- 5Back-substitute and check in bothPut the number into either original to get the other variable. Check the pair in both.
Multiply one equation: x + 2y = 7 and 3x − y = 7
The y coefficients are 2 and −1. Multiply the second equation by 2 (both sides): 6x − 2y = 14. Now 2y and −2y are opposites. Add: 7x = 21, so x = 3. Back-substitute: 3 + 2y = 7, so y = 2. Check: 3 + 4 = 7; 9 − 2 = 7. (3, 2).
Multiply both: 2x + 3y = 8 and 3x + 2y = 7
To cancel x, make the coefficients 6 and −6: multiply the first by 3 (6x + 9y = 24) and the second by −2 (−6x − 4y = −14). Add: 5y = 10, so y = 2. Back-substitute: 2x + 6 = 8, so x = 1. Check: 2 + 6 = 8; 3 + 4 = 7. (1, 2).
Subtracting insteadIf the coefficients are already equal (3y and 3y), subtract the equations. Or multiply one by −1 and add; same thing.
Multiply the whole equationBoth sides, every term. That is the multiply-both-sides move from Unit 1, and it keeps the equation true.
Back-substitute into eitherBoth originals are true at the solution, so either one gives the other variable.