Make one letter disappear
Make the coefficients equal, then add or subtract.
Take one from the other
Look at 4x + 6y = 24 and 9x + 6y = 39. The y-terms are the same size.
What happens to y if we subtract one equation from the other?
What this lesson covers
The idea
Multiply the equations by suitable non-zero constants to make the coefficients of one variable numerically equal, add or subtract to eliminate that variable, solve the one-variable equation, and substitute back to find the other variable.
Take one from the other
Look at 4x + 6y = 24 and 9x + 6y = 39. The y-terms are the same size.
What happens to y if we subtract one equation from the other?
- y disappears
- y doubles
- Nothing changes
Make a letter disappear
Pick a letter to get rid of. Multiply the equations so its coefficients are numerically equal, then subtract or add. Do it once for x and once for y.
Elimination, step by step
Step 1: multiply the equations by suitable non-zero numbers so that the coefficients of one letter are numerically equal. Step 2: subtract or add so that this letter is eliminated. Step 3: solve the equation in one letter. Step 4: put the value back to find the other letter.
The book does 9x − 4y = 2000 and 7x − 3y = 2000: multiply by 3 and by 4: 27x − 12y = 6000 and 28x − 12y = 8000 subtract: x = 2000 then 9(2000) − 4y = 2000, so y = 4000
If no letter is left after Step 2, read the statement: true means infinitely many solutions, false means no solution.
Notes
Multiply to make the coefficients of one letter numerically equal, then subtract or add to eliminate it, solve, and put the value back.
Check yourself
To get rid of y in 3x + 2y = 7 and 2x + 5y = 12, which multipliers make the y-coefficients equal?
x + 3y = 7 and 2x − 3y = 5. The y-terms are 3y and −3y. What should we do?
The book gets 27x − 12y = 6000 and 28x − 12y = 8000. Subtract the first from the second. What is x?
Answer: 2000
(28x − 27x) − (12y − 12y) = 8000 − 6000, so x = 2000.
Solve 3x + 2y = 11 and x − 2y = 1. Add them to find x = 3. What is y?
Answer: 1
3 − 2y = 1 gives y = 1. Check in the first equation: 3 × 3 + 2 × 1 = 11 ✓.
- Multiply the first by 5 and the second by 2 — correct. Yes! Both y-coefficients become 10.
- Multiply the first by 2 and the second by 5. The y-coefficients would be 4 and 25: not equal.
- Multiply both by 3. The y-coefficients would be 6 and 15: not equal.
- Add the equations — correct. Yes! 3y + (−3y) = 0, so y vanishes and 3x = 12.
- Subtract the equations. Subtracting gives 3y − (−3y) = 6y, so y stays.
- Multiply both by 3. Not needed: the y-terms are already the same size.