Scale the equations, then take one away
Bus fares: three adults and two children pay ₹370. One of each pays ₹140. Let x be the adult fare and y the child fare. Scale, then subtract.
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3x + 2y = 370 and x + y = 140.3x + 2y = 370andx + y = 140.- Scale the second by 2:
2x + 2y = 280. - Subtract. The y-terms cancel.
x = 90falls out. Then y = 50.
The rule
Rule: Scale a whole equation so one letter has matching coefficients. Then subtract. That letter falls out.
To scale an equation is to multiply every part by one number. A letter that falls out has gone from the equation. Removing a letter this way is called elimination.
Scale the second equation by 2. Its y-coefficient now matches: 2x + 2y = 280. Subtract it from 3x + 2y = 370. The left sides give 3x − 2x = x, and the y-terms cancel. The right sides give 370 − 280 = 90. So x = 90. Then y = 140 − 90 = 50.
| Equation | At (90, 50) | Claim says |
|---|---|---|
3x + 2y = 370 | 270 + 100 = 370 | 370 |
x + y = 140 | 90 + 50 = 140 | 140 |
An adult ticket costs ₹90. A child ticket costs ₹50.
Either letter can be removed first. Scale the second equation by 3 instead: 3x + 3y = 420. Subtract it from 3x + 2y = 370. The x-terms cancel and leave −y = −50. So y = 50 again.
Watch out!
Scale every part, including the number on the right. 2x + 2y = 140 is wrong. Twice 140 is 280.