Meera plotted both equations of a pair on the same grid and got lines ℓ₁ and ℓ₂ shown. What can she conclude about the pair?

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

AIt has infinitely many solutions
BIts solution is $x = 0, y = 1$
CIts solution is $x = 4, y = 0$
DIt has no solution

Step-by-Step Solution

ℓ_{1}

and

ℓ_{2}

rise by 1 for every 2 steps right and never meet: they are parallel. Parallel lines have no common point, so the pair is inconsistent and has no solution.

Answer: D) It has 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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