If a pair of linear equations has a1/a2 = b1/b2 = c1/c2, the lines are:

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations

ACoincident (infinitely many solutions)
BIntersecting (one solution)
CParallel (no solution)
DPerpendicular

Step-by-Step Solution

When all three coefficient ratios are equal, the two equations represent the SAME line, so they coincide and have infinitely many solutions.

Answer: A) Coincident (infinitely many solutions)

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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