- A(4, 2)
- B(1, 0)
- C(2, 4)
- D(3, 8)
Step-by-Step Solution
From the first line . Substituting: → → and . Check: ✓ and ✓. (4, 2) is the answer with the coordinates written in the wrong order; (1, 0) lies on the first line only.
Answer: C) (2, 4)
Sonal draws the lines and on graph paper. Reading her graph, at which point do the two lines meet?
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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