Two simultaneous equations given by y = 1/2 + x and y = x - 1/2 have:
Options:
(A) a unique solution
(B) infinitely many solutions
(C) no solution
(D) a finite number of multiple solutions
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2011 Q.5)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (1 Mark, MCQ)
- **Topic**: Analytical & Spatial Aptitude (Spatial Reasoning & Paper Folding)
- **Organizing Institute**: IIT Madras
- **Problem Statement**:
> Two simultaneous equations given by $y = \pi + x$ and $y = x - \pi$ have:
**2. Mathematical Derivation & Physical Analysis**:
1. **System of Linear Equations**:
Given lines: $y = \pi + x \implies y - x = \pi$ and $y = x - \pi \implies y - x = -\pi$.
2. **Geometric & Algebraic Analysis**:
Both equations have the identical slope $m = 1$, but distinct y-intercepts ($c_1 = \pi, c_2 = -\pi$).
Subtracting the two equations yields $0 = 2\pi$, which is an impossible contradiction ($2\pi \ne 0$).
Geometrically, these are two distinct parallel lines in the Euclidean plane and thus never intersect.
3. **Conclusion**:
The system has **no solution** (**Option C**).
**3. Comprehensive Options Analysis**:
- **(A)** *"a unique solution"*: ❌ **INCORRECT**.
- **(B)** *"infinitely many solutions"*: ❌ **INCORRECT**.
- **(C)** *"no solution"*: ✅ **CORRECT**.
- **(D)** *"a finite number of multiple solutions"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2011 (IIT Madras) master answer key with 100% precision.
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