- A6
- B4
- C2
- D3
Step-by-Step Solution
Infinite solutions require 2/4 = p/6 = 4/8. 2/4 = 1/2, so p/6 = 1/2 → p = 3. (Check 4/8 = 1/2 ✓.)
Answer: D) 3
For what value of will the system and have infinitely many solutions?
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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