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The two lines drawn for a pair of equations are found to have the SAME slope but DIFFERENT -intercepts. The pair is:

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • AConsistent with a unique solution
  • BConsistent with infinitely many solutions
  • CInconsistent, with no solution
  • DConsistent with exactly two solutions

Step-by-Step Solution

Equal slopes make the lines parallel; different -intercepts keep them apart, so they never meet and there is no common point — the pair is inconsistent. Equal slopes AND the same intercept would give one line (infinitely many solutions), and two straight lines can never cross at exactly two points.
Answer: C) Inconsistent, with no solution

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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