The variables x, y, and z satisfy x gt 0, y gt 0, and z gt y, and are related...

1. TRANSLATE the problem information

• Given equation: \(\frac{8}{\mathrm{x}} - \frac{4}{\mathrm{y}} = -\frac{4}{\mathrm{z}}\)
• Constraints: \(\mathrm{x} \gt 0\), \(\mathrm{y} \gt 0\), \(\mathrm{z} \gt \mathrm{y}\)
• Goal: Find an expression for x

2. INFER the solution strategy

• Since we need x and it appears in a fraction, we should isolate the term containing x first
• Rearranging to get \(\frac{8}{\mathrm{x}}\) by itself will make solving for x more straightforward

3. SIMPLIFY by rearranging the equation

• Move \(\frac{4}{\mathrm{y}}\) to the right side:
  \(\frac{8}{\mathrm{x}} = \frac{4}{\mathrm{y}} - \frac{4}{\mathrm{z}}\)

• Factor out 4:
  \(\frac{8}{\mathrm{x}} = 4\left(\frac{1}{\mathrm{y}} - \frac{1}{\mathrm{z}}\right)\)

4. SIMPLIFY the fraction subtraction

• Find common denominator for \(\frac{1}{\mathrm{y}} - \frac{1}{\mathrm{z}}\):
  \(\frac{1}{\mathrm{y}} - \frac{1}{\mathrm{z}} = \frac{\mathrm{z}}{\mathrm{yz}} - \frac{\mathrm{y}}{\mathrm{yz}} = \frac{\mathrm{z}-\mathrm{y}}{\mathrm{yz}}\)

• Substitute back:
  \(\frac{8}{\mathrm{x}} = \frac{4(\mathrm{z}-\mathrm{y})}{\mathrm{yz}}\)

5. SIMPLIFY to solve for x

• Take reciprocal of both sides:
  \(\frac{\mathrm{x}}{8} = \frac{\mathrm{yz}}{4(\mathrm{z}-\mathrm{y})}\)

• Multiply both sides by 8:
  \(\mathrm{x} = 8 \cdot \frac{\mathrm{yz}}{4(\mathrm{z}-\mathrm{y})} = \frac{2\mathrm{yz}}{\mathrm{z}-\mathrm{y}}\)

Answer: D. \(\frac{2\mathrm{yz}}{\mathrm{z}-\mathrm{y}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make errors when subtracting fractions with different denominators or when manipulating the final algebraic expression.

When finding \(\frac{1}{\mathrm{y}} - \frac{1}{\mathrm{z}}\), they might incorrectly write \(\frac{1-1}{\mathrm{y}-\mathrm{z}} = \frac{0}{\mathrm{y}-\mathrm{z}}\) instead of properly using the common denominator yz. This leads to \(\frac{8}{\mathrm{x}} = 0\), which would mean x is undefined. This causes confusion and typically leads to random guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning about solution strategy: Students attempt to clear denominators by multiplying through by xyz immediately, creating a more complex equation to manipulate.

Starting with \(\mathrm{xyz}\left(\frac{8}{\mathrm{x}} - \frac{4}{\mathrm{y}}\right) = \mathrm{xyz}\left(-\frac{4}{\mathrm{z}}\right)\) gives \(8\mathrm{yz} - 4\mathrm{xz} = -4\mathrm{xy}\), which leads to a messier algebraic manipulation. Students often make sign errors or struggle to isolate x from this form, potentially selecting Choice C (\(\frac{2\mathrm{yz}}{\mathrm{y}-\mathrm{z}}\)) by getting the sign wrong in the denominator.

The Bottom Line:

Success on this problem requires systematic algebraic manipulation skills, particularly with rational expressions. The key insight is recognizing that isolating the x-term first simplifies the overall solution process.