Checking a pair twice
A pair like (3, 2) means x = 3 and y = 2. A pair can fit one equation. Can it fit two equations? Level means both sides are equal.
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2x + 3y = 12 and x − y = 1. Step 1 of 4
- Candidate (3, 2), for
2x + 3y = 12andx − y = 1. - First claim: 2·3 + 3·2 = 12. Level.
- Second claim: 3 − 2 = 1. Level too.
- Both level, so (3, 2) solves both equations.
The rule
Only if it makes both equations level. A pair of equations means two equations taken together. To satisfy an equation is to make its two sides equal. A solution is a pair of numbers, like (3, 2).
Rule: A solution of a pair of equations must satisfy both equations. Not just one.
One equation in two letters accepts many pairs. 2x + 3y = 12 accepts (6, 0), (0, 4) and (3, 2). It even accepts (−3, 6). But a puff cannot cost −₹3.
| Candidate | 2x + 3y | Claim says | Level? |
|---|---|---|---|
| (6, 0) | 12 | 12 | yes |
| (0, 4) | 12 | 12 | yes |
| (3, 2) | 12 | 12 | yes |
A pair of linear equations is two such claims side by side.
An ordered pair (x, y) names one exact point. A solution of the equations is a point where both claims agree.