Which of the following pairs of equations is consistent with a UNIQUE solution?

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

A $x + 2 y = 4$ and $3 x + y = 7$
B $x + 2 y = 4$ and $2 x + 4 y = 8$
C $x + 2 y = 4$ and $2 x + 4 y = 12$
D $x + 2 y = 4$ and $x + 2 y = 9$

Step-by-Step Solution

For

x + 2 y = 4

and

3 x + y = 7

: ratios 1/3 ≠ 2/1, so unique solution. The second pair is coincident (infinite), the third and fourth are parallel (none).

Answer: A)

x + 2 y = 4

and

3 x + y = 7

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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