If $$x$$ and $$y$$ satisfy the equations $$|x| + x + y = 15$$ and $$x + |y| - y = 20$$, then $$(x - y)$$ equals

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Accepted Answer

The problem involves solving two equations involving absolute values to determine $$x - y$$. Here's the step-by-step solution:

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1742462463/aptidude_questions/gpgpjtiw4mdgfzbcjsch.png)

### **Given Equations:**
1. $$|x| + x + y = 15$$  
2. $$x + |y| - y = 20$$

Absolute value equations depend on the signs of $$x$$ and $$y$$. We analyze each quadrant:

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### **Case 1: First Quadrant ($$x > 0, y > 0$$)**

- Substitute $$|x| = x$$ and $$|y| = y$$:
  $$
  x + x + y = 15 \quad \Rightarrow \quad 2x + y = 15
  $$
  $$
  x + y - y = 20 \quad \Rightarrow \quad x = 20
  $$
- Substitute $$x = 20$$ into $$2x + y = 15$$:
  $$
  2(20) + y = 15 \quad \Rightarrow \quad y = -25
  $$
- Since $$y > 0$$ in the first quadrant, this case is invalid.

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### **Case 2: Fourth Quadrant ($$x > 0, y <   0$$)**

- Substitute $$|x| = x$$ and $$|y| = -y$$:
   $$
   x + x + y = 15 \quad \Rightarrow \quad 2x + y = 15
   $$
   $$
   x - (-y) - (-y) = 20 \quad \Rightarrow \quad x - 2y = 20
   $$
- Solve these two equations simultaneously:
   - From $$2x + y = 15$$:
     $$
     y = (15 - 2x)
     $$
   - Substitute into $$x - (2y) = (20)$$:
     $$
     x - (2(15 - (2x))) = (20)
     $$
     Simplify:
     $$
     x - (30 - (4x)) = (20)
     $$
     Combine terms:
     $$
     x + (4x) - (30) = (20)
     $$
     Solve for $$x$$:
     $$
     (5x) = (50) \quad \Rightarrow \quad x = (10)
     $$
   - Substitute $$x=10$$ into $$y=(15-(2)(10))$$:
     $$
     y=(-5)
     $$

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### **Calculate $$x-y$$:**

- Subtract values:
   $$
   x-y=(10)-(-5)=10+5=15
   $$

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### Final Answer:
The value of $$x-y$$ is **15**.  
Correct option: **B**

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Answer from Perplexity: pplx.ai/share

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