CAT 2024 Slot 2 QA Question 14

Multiple choice (+3 / −1) · Algebra · Modulus · Try it, then check the answer and solution below.

If x and y are real numbers such that

|x| + x + y = 15

and

x + |y| - y = 20 , then

(x - y)$ equals

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: C

Answer and solution

Answer: C) 15

📌 Core Concept

The problem involves solving equations with absolute values. The key is to consider different cases based on the signs of the variables.

🔢 Step-by-Step Solution

First Equation:

|x| + x + y = 15

Case 1: x ≥ 0$

2x + y = 15 \quad \text{(Equation 1a)}

Case 2: x < 0$

y = 15 \quad \text{(Equation 1b)}

Second Equation:

x + |y| - y = 20

Subcase 1a: y ≥ 0$

x = 20 \quad \text{(Contradictsy \geq 0)}

Subcase 1b: y < 0$

x - 2y = 20 \quad \text{(Equation 2a)}

Solve Equations 1a and 2a:

\begin{cases} 2x + y = 15 \\ x - 2y = 20 \end{cases}

Multiply Equation 1a by 2:

4x + 2y = 30

Add to Equation 2a:

\[ 5x = 50 \impl

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