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Assertion (A): The pair and has infinitely many solutions. Reason (R): The two equations represent the same straight line. 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 3/6 = 2/4 = 5/10 = 1/2, so the lines coincide → infinitely many solutions (A true). They are the same line, which is exactly why there are infinite solutions (R explains A).
Answer: A) Both A and R are true and R is the correct explanation of A

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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