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Assertion (A): The pair and has no solution. Reason (R): For this pair, a₁/a₂ = ₁/₂ = ₁/₂. Choose the correct option.

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • ABoth A and R are true and R is the correct explanation of A
  • BBoth A and R are true but R is not the correct explanation of A
  • CA is true but R is false
  • DA is false but R is true

Step-by-Step Solution

Ratios: 2/4 = 1/2 and 3/6 = 1/2, but 9/7 ≠ 1/2. So a₁/a₂ = b₁/b₂ ≠ c₁/c₂ — parallel lines, no solution, and A is TRUE. R claims all three ratios are equal, which is the condition for infinitely many solutions, so R is FALSE (and it would even contradict A).
Answer: C) A is true but R is false

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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