Solving Absolute Value Equations: A Math Guide

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Hey math enthusiasts! Today, we're diving into a problem that involves absolute values and integer solutions. Specifically, we're going to solve for x and y given two equations: |x| = 32 + x - y and |y| = 24 - x - y. Our ultimate goal? To find the value of x + y. Let's break this down step-by-step to make sure we understand it.

Understanding Absolute Values and Equations

First off, let's refresh our memory on what absolute value means. The absolute value of a number is its distance from zero on the number line. This means it's always a non-negative value. For example, |3| = 3 and |-3| = 3. Now, when we have equations with absolute values, we need to consider different cases depending on whether the expressions inside the absolute value are positive or negative. The keywords here are absolute value, equations, and integer solutions. We are solving an equation and must consider the absolute value, and our solution has to be an integer. This means we cannot find solutions in the format of fractions and decimals. We need to satisfy the criteria of the given question. Let’s look at the given equations, |x| = 32 + x - y and |y| = 24 - x - y. To solve this, we will first need to find the value of x, and also y, using these two equations. Then add them to find the value of x + y.

Breaking Down the Absolute Value Equations

Let’s start with the first equation, |x| = 32 + x - y. Because of the absolute value, we need to consider two cases:

  1. Case 1: x ≥ 0 If x is non-negative, then |x| = x. So, our equation becomes x = 32 + x - y. Simplifying, we get 0 = 32 - y, which means y = 32.

  2. Case 2: x < 0 If x is negative, then |x| = -x. Our equation becomes -x = 32 + x - y. Rearranging, we get y = 32 + 2x.

Now, let’s move on to the second equation, |y| = 24 - x - y, and similarly consider two cases:

  1. Case 1: y ≥ 0 If y is non-negative, then |y| = y. Our equation becomes y = 24 - x - y. Rearranging, we get 2y = 24 - x, or x = 24 - 2y.

  2. Case 2: y < 0 If y is negative, then |y| = -y. Our equation becomes -y = 24 - x - y. Simplifying, we get 0 = 24 - x, which means x = 24.

Combining the Cases to Solve for x and y

Now, we need to combine these cases to find the values of x and y that satisfy both equations. This is where it gets a little bit more interesting, guys. Stick with me! We have to find combinations of x and y that work in both equations.

Subcase 1: x ≥ 0 and y ≥ 0

From Case 1 of Equation 1, we have y = 32. From Case 1 of Equation 2, we have x = 24 - 2y. Substituting y = 32 into the second equation, we get x = 24 - 2(32) = 24 - 64 = -40. However, this contradicts our assumption that x ≥ 0. So, this combination doesn’t work.

Subcase 2: x ≥ 0 and y < 0

From Case 1 of Equation 1, we have y = 32. But, this contradicts our assumption that y < 0. So, this combination doesn’t work.

Subcase 3: x < 0 and y ≥ 0

From Case 2 of Equation 1, we have y = 32 + 2x. From Case 1 of Equation 2, we have x = 24 - 2y. Substituting the first equation into the second, we get x = 24 - 2(32 + 2x). Simplifying, x = 24 - 64 - 4x, which gives us 5x = -40, so x = -8. Now, substituting x = -8 back into y = 32 + 2x, we get y = 32 + 2(-8) = 32 - 16 = 16. Since x = -8 and y = 16 satisfy our initial assumptions ( x < 0 and y ≥ 0), this is a valid solution.

Subcase 4: x < 0 and y < 0

From Case 2 of Equation 1, we have y = 32 + 2x. From Case 2 of Equation 2, we have x = 24. Substituting x = 24 into the first equation, we get y = 32 + 2(24) = 32 + 48 = 80. However, this contradicts our assumption that y < 0. So, this combination doesn’t work.

Finding x + y

From the above, the only solution that satisfies both equations is x = -8 and y = 16. Therefore, x + y = -8 + 16 = 8. So, the answer to our question is 8! Remember, the keywords were absolute value, equations, and integer solutions. We applied the definition of absolute values to write the equations. We considered all cases. We also carefully checked the validity of our solutions. We considered all subcases. We have found the integer value of x and y that satisfy the given conditions.

Summary and Key Takeaways

Alright, let's recap what we've learned:

  • Absolute Values: The absolute value of a number is its distance from zero. Always consider both positive and negative cases when dealing with absolute values.
  • Solving Equations: Break down the equations into different cases based on the sign of the expressions inside the absolute values.
  • Combining Solutions: Carefully combine the solutions from each case, ensuring that they satisfy all original conditions and assumptions.
  • x + y: Once you find the correct values of x and y, simply add them together to get your final answer.

This method can be applied to many other math problems involving absolute values. The key is to be methodical and break the problem down into manageable steps. Keep practicing, and you'll get the hang of it, folks!

Practical Tips and Tricks

To make this process even smoother, here are some practical tips:

  • Organize Your Work: Write down each case and subcase clearly. This will help you avoid mistakes and keep track of your progress.
  • Check Your Assumptions: Always verify that your solutions match the initial assumptions for each case.
  • Practice, Practice, Practice: The more you solve these types of problems, the more comfortable and confident you'll become. Try variations of the problem, and you will understand more deeply.

Remember, mastering math is about understanding the principles and applying them creatively. So, keep exploring, keep questioning, and keep having fun! If you follow all these steps, you should have no problem solving the problem. The keywords of absolute value, equations, and integer solutions are the most important part of this entire question, and you will not have any problems if you follow the procedure. Keep up the amazing work! If you have any further questions, feel free to ask me. I am always happy to help!