Solving Absolute Value Equations: |x| + y = 10 and x + |y| = 6

Introduction

In this blog post, we’ll solve a mathematical problem involving absolute values and linear equations. The problem is as follows:

Problem Statement:
Given the equations |x| + y = 10 and x + |y| = 6, find the value of x + y.

Let’s break down the solution step-by-step.

Step 1: Understanding the Equations

We are given two equations involving absolute values:

1. |x| + y = 10
2. x + |y| = 6

The absolute value function affects the equations differently based on the signs of x and y. We’ll consider different cases based on the signs of x and y.

Step 2: Case Analysis

To solve these equations, we analyze four possible cases based on the signs of x and y.

Case 1: x ≥ 0 and y ≥ 0

In this case, |x| = x and |y| = y. The equations become:

1. x + y = 10
2. x + y = 6

Clearly, these two equations contradict each other. Thus, there is no solution in this case.

Case 2: x ≥ 0 and y < 0

Here, |x| = x and |y| = -y. The equations are:

1. x + y = 10
2. x – y = 6

Solving these:

Adding the two equations: 2x = 16 → x = 8
Substituting x = 8 into x + y = 10: 8 + y = 10 → y = 2

However, this gives y = 2, which contradicts our assumption that y < 0. Hence, there is no solution in this case.

Case 3: x < 0 and y ≥ 0

In this case, |x| = -x and |y| = y. The equations become:

1. -x + y = 10
2. x + y = 6

Solving these:

Subtracting the second equation from the first: -2x = 4 → x = -2
Substituting x = -2 into x + y = 6: -2 + y = 6 → y = 8

So, we have x = -2 and y = 8. Both values satisfy our conditions x < 0 and y ≥ 0.

Case 4: x < 0 and y < 0

In this scenario, |x| = -x and |y| = -y. The equations are:

1. -x + y = 10
2. x – y = 6

Solving these:

Adding the two equations: -2x = 16 → x = -8
Substituting x = -8 into x – y = 6: -8 – y = 6 → -y = 14 → y = -14

However, this gives y = -14, which contradicts our assumption that y < 0. Hence, there is no solution in this case.

Step 3: Conclusion

The only valid solution comes from Case 3, where x = -2 and y = 8. Thus:

x + y = -2 + 8 = 6

Therefore, the value of x + y is 6.

Final Thoughts

This problem highlights the importance of considering all possible cases when dealing with absolute values. By carefully analyzing each scenario, we can determine the correct solution.

Feel free to comment below if you have any questions or if you’d like to see more examples of solving equations with absolute values!