Question

Which of the following equations best describes the graph given below?

• A. |x+y|-|x-y|=6
• B. |x-y|+|x+y|=10
• C. |x|-|y|=6
• D. |x+y|-|xy|=0

Answer 1

The Correct Option is C

In this problem, we are given a choice of equations and need to determine which describes a specific graph. The correct answer is \(|x| - |y| = 6\). Let's explore why this is the case and examine why the other options do not fit correctly either.

To understand the equation \(|x| - |y| = 6\) and its graphical representation, consider the following steps: 

1. The equation \(|x| - |y| = 6\) implies that the absolute values of \(x\) and \(y\) have a difference of 6 units.
2. This means the graph consists of two lines, based on the conditions:
   • \(x \geq y + 6\) where \(x \geq 0\) and \(y \geq 0\).
   • \(x \geq -y + 6\) where \(x \geq 0\) and \(y \leq 0\).
   • \(-x \geq y + 6\) where \(x \leq 0\) and \(y \geq 0\).
   • \(-x \geq -y + 6\) where \(x \leq 0\) and \(y \leq 0\).
3. These linear segments form the boundaries of a region where the absolute difference remains constant.

Let's consider why the other options are not correct:

• \(|x+y| - |x-y| = 6\): This describes two parallel lines where the distance between them is affected by the simultaneous values of \(x\) and \(y\), not fitting the specific conditions of the given equation.
• \(|x-y| + |x+y| = 10\): This expression sums two absolute differences, resulting in a diamond shape if graphed. The boundaries are significantly different from the equation representing uniform difference like \(|x| - |y| = 6\).
• \(|x+y| - |xy| = 0\): This relationship involves multiplication and would produce a more complex curve than simple linear boundaries.

Based on the explanations above, the equation \(|x|-|y|=6\) best describes the graph because it accounts for a linear boundary through consistent absolute differences.