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For which values of k does the pair kx + and have a UNIQUE solution?

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • A
  • Bk ≠ 6
  • Ck ≠ 3/2
  • D

Step-by-Step Solution

A unique solution needs a₁/a₂ ≠ b₁/b₂, i.e. ≠ 2/1, so ≠ 6. At the ratios of - and -coefficients both equal 2 while c₁/c₂ = 5, so the lines are parallel and there is no solution. Writing the ratio upside down as gives the wrong value 3/2.
Answer: B) k ≠ 6

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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