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

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: $x - y = 15.$