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The pair and has infinitely many solutions. Find .

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • A5
  • B−8
  • C7
  • D3

Step-by-Step Solution

Infinitely many solutions need 2/(k − 1) = 3/(k + 2) = 7/(3k). From the first two: 2(k + 2) = 3(k − 1) → 2k + 4 = 3k − 3 → k = 7. Check with the third: 2/6 = 3/9 = 7/21 = 1/3 ✓, so the second equation is exactly 3 times the first. Cross-multiplying the wrong way as 2(k − 1) = 3(k + 2) gives the trap value −8.
Answer: C) 7

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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