Both roads, one answer
Two badges and a flag cost ₹90. Five badges and three flags cost ₹235. Let x be a badge's price and y a flag's price. That gives 2x + y = 90 and 5x + 3y = 235. Walk both roads. A road is one method.
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2x + y = 90 and 5x + 3y = 235.- Start:
2x + y = 90and5x + 3y = 235. - Substitution road: y = 90 − 2x, then x = 35.
- Elimination road: scale by 3, subtract, x = 35 too.
- Both roads land on (35, 20).
The rule
Rule: Substitution and elimination both describe the same two lines. The two roads cannot land on two different answers.
The first equation gives y = 90 − 2x. Put that in the second equation: 5x + 3(90 − 2x) = 235. That is 270 − x = 235, so x = 35. Then y = 90 − 70 = 20. For elimination, scale the first equation by 3: 6x + 3y = 270. Subtract the second equation. The y-terms cancel, and 6x − 5x = x while 270 − 235 = 35. So x = 35 again.
| Equation | At (35, 20) | Claim says |
|---|---|---|
2x + y = 90 | 70 + 20 = 90 | 90 |
5x + 3y = 235 | 175 + 60 = 235 | 235 |
Badges cost ₹35 each. Flags cost ₹20 each.
A letter with no number in front makes substitution easy. Coefficients that are easy to match make elimination easy. Here both roads are easy.