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On the same graph paper, Aditya plots and . Which statement correctly describes his two lines?

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • AThey cross at exactly one point
  • BThey lie exactly on top of each other
  • CThey never meet, however far they are extended
  • DThey cut the -axis at the same point

Step-by-Step Solution

a₁/a₂ = 1/2 and b₁/b₂ = , but c₁/c₂ = 3/9 = 1/3. Equal coefficient ratios with a different constant ratio means parallel distinct lines, so they never meet. They also cut the -axis at (3, 0) and (4.5, 0) — different points, so the last option is false too.
Answer: C) They never meet, however far they are extended

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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