Question

If \( x \) and \( y \) satisfy the equations
\(|x| + x + y = 15 \quad \text{(1)},\)
\(x + |y| - y = 20 \quad \text{(2)}.\)
Find the value of \( x - y \).

Answer 1

Correct Answer: 15

Step 1: Analyze the first equation \( |x| + x + y = 15 \)}

The behavior of \( |x| \) depends on the sign of \( x \):

• If \( x \geq 0 \), then \( |x| = x \). Substituting this into Equation (1):
  \(x + x + y = 15 \implies 2x + y = 15. \quad \text{(3)}\)
• If \(x < 0\), then \( |x| = -x \). Substituting this into Equation (1):
     \(-x + x + y = 15 \implies y = 15. \quad \text{(4)}\)
   

Step 2: Analyze the second equation \( x + |y| - y = 20 \)}

The behavior of \( |y| \) depends on the sign of \( y \):

• If \( y \geq 0 \), then \( |y| = y \). Substituting this into Equation (2):
   \(x + y - y = 20 \implies x = 20. \quad \text{(5)}\)
• If \(y < 0\), then \( |y| = -y \). Substituting this into Equation (2):
    \(x - y - y = 20 \implies x - 2y = 20. \quad \text{(6)}\)

Step 3: Solve the equations for different cases

Case 1: \( x \geq 0 \) and \( y \geq 0 \)

From Equation (3):
\(2x + y = 15.\)

From Equation (5):
\(x = 20.\)

Substitute \( x = 20 \) into \( 2x + y = 15 \):
\(2(20) + y = 15 \implies 40 + y = 15 \implies y = -25.\)

This violates the assumption \( y \geq 0 \). Thus, this case is not valid.

Case 2: \( x \geq 0 \) and \(y < 0\)

From Equation (3):
\(2x + y = 15.\)

From Equation (6):
\(x - 2y = 20.\)

Solve these two equations simultaneously:
1. From \( 2x + y = 15 \), express \( y \) in terms of \( x \):
\(y = 15 - 2x. \quad \text{(7)}\)

2. Substitute \( y = 15 - 2x \) into \( x - 2y = 20 \):
\(x - 2(15 - 2x) = 20.\)

Simplify:
\(x - 30 + 4x = 20 \implies 5x - 30 = 20 \implies 5x = 50 \implies x = 10.\)

Substitute \( x = 10 \) into \( y = 15 - 2x \):
\(y = 15 - 2(10) = 15 - 20 = -5.\)

 Step 4: Calculate \( x - y \) 
From the above, \( x = 10 \) and \( y = -5 \). 
Thus: \(x - y = 10 - (-5) = 10 + 5 = 15.\)
Final Answer: \(x - y = 15.\)