The equation 12/t = 12/x - 12/y - 12/z relates the positive variables t, x, y, and z. Which of...

1. TRANSLATE the problem information

• Given equation: \(\frac{12}{\mathrm{t}} = \frac{12}{\mathrm{x}} - \frac{12}{\mathrm{y}} - \frac{12}{\mathrm{z}}\)
• Goal: Find an expression equivalent to x from the answer choices

2. INFER the strategic approach

• Notice that 12 appears in every term - factor it out first to simplify
• After factoring, work with the simpler equation to isolate x

3. SIMPLIFY by factoring out the common term

• Factor 12 from the right side: \(\frac{12}{\mathrm{t}} = 12\left(\frac{1}{\mathrm{x}} - \frac{1}{\mathrm{y}} - \frac{1}{\mathrm{z}}\right)\)
• Divide both sides by 12: \(\frac{1}{\mathrm{t}} = \frac{1}{\mathrm{x}} - \frac{1}{\mathrm{y}} - \frac{1}{\mathrm{z}}\)

4. SIMPLIFY to isolate the term containing x

• Rearrange to solve for \(\frac{1}{\mathrm{x}}\): \(\frac{1}{\mathrm{x}} = \frac{1}{\mathrm{t}} + \frac{1}{\mathrm{y}} + \frac{1}{\mathrm{z}}\)
• Now we need to add these three fractions

5. SIMPLIFY by finding a common denominator

• Common denominator for t, y, and z is tyz
• \(\frac{1}{\mathrm{x}} = \frac{\mathrm{yz}}{\mathrm{tyz}} + \frac{\mathrm{tz}}{\mathrm{tyz}} + \frac{\mathrm{ty}}{\mathrm{tyz}}\)
• \(\frac{1}{\mathrm{x}} = \frac{\mathrm{yz + tz + ty}}{\mathrm{tyz}}\)

6. SIMPLIFY by taking the reciprocal

• If \(\frac{1}{\mathrm{x}} = \frac{\mathrm{yz + tz + ty}}{\mathrm{tyz}}\), then \(\mathrm{x} = \frac{\mathrm{tyz}}{\mathrm{yz + tz + ty}}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when combining fractions or finding common denominators. They might incorrectly add the numerators without proper common denominators, or make sign errors when rearranging terms. This leads to expressions that don't match any of the given choices, causing confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about reciprocals: Students correctly get to \(\frac{1}{\mathrm{x}} = \frac{\mathrm{yz + tz + ty}}{\mathrm{tyz}}\) but then struggle with taking the reciprocal properly. They might flip only part of the fraction or confuse which terms go in the numerator vs. denominator. This may lead them to select Choice C (\(\frac{\mathrm{xyz}}{\mathrm{yz + xz + xy}}\)) where they've incorrectly included x in both the numerator and denominator.

The Bottom Line:

This problem tests systematic algebraic manipulation skills more than conceptual understanding. Success requires careful attention to each algebraic step, especially when working with multiple fractions and reciprocals.