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The pair and has NO solution. Find .

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • A12
  • B1.5
  • C−3
  • D3

Step-by-Step Solution

No solution needs a₁/a₂ = b₁/b₂ ≠ c₁/c₂, so 4/2 = 6/k → 2k = 6 → k = 3. Check the constants: c₁/c₂ = 11/7 ≠ 2 ✓, so the lines really are parallel. The value 12 comes from inverting one ratio (writing 4/2 = k/6), and 1.5 from inverting the other.
Answer: D) 3

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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