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Is (3, −2) a solution of the pair and ?

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • ANo — it satisfies only the first equation
  • BYes — it satisfies both equations
  • CNo — it satisfies only the second equation
  • DNo — it satisfies neither equation

Step-by-Step Solution

First: 2(3) + 5(−2) = 6 − 10 = −4 ✓. Second: 4(3) − 3(−2) = 12 + 6 = 18 ✓. A pair is a solution only when BOTH equations are satisfied, and here both are. Students who stop after checking one equation pick one of the 'only' options.
Answer: B) Yes — it satisfies both equations

Key idea: Three geometric possibilities

Two lines in a plane can:

1. **Intersect at a single point** → exactly ONE solution. (Independent, consistent.)
2. **Be parallel and distinct** → NO solution. (Inconsistent.)
3. **Coincide (be the same line)** → INFINITELY many solutions. (Dependent, consistent.)

These three cases are decided by comparing the coefficient ratios:

| Condition on a₁/a₂, b₁/b₂, c₁/c₂ | Type | # solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Lines intersect | One |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel | None |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident | Infinitely many |

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