- AIntersect at one point
- BAre parallel
- CAre coincident
- DDo not exist
Step-by-Step Solution
a1/a2 = 3/2, b1/b2 = 2/(−3) = −2/3. Since 3/2 ≠ −2/3, the lines intersect at exactly one point.
Answer: A) Intersect at one point
For the pair and , the lines:
Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
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 |
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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