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For the pair and , the two lines are:

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • AIntersecting — the system has a unique solution
  • BCoincident — the system has infinitely many solutions
  • CPerpendicular — the system has a unique solution
  • DParallel — the system is inconsistent

Step-by-Step Solution

a₁/a₂ = 5/10 = 1/2 and b₁/b₂ = (−4)/(−8) = 1/2, but c₁/c₂ = 8/20 = 2/5. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂ the lines are parallel and the system has no solution. Stopping after the first two equal ratios is what makes students say 'coincident'.
Answer: D) Parallel — the system is inconsistent

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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