b - 72 = x/y The given equation relates the positive numbers b, x, and y. Which equation correctly expresses...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{b - 72 = \frac{x}{y}}\)
• Goal: Express x in terms of b and y

2. INFER the solution strategy

• Since x appears in the numerator of a fraction \(\mathrm{\left(\frac{x}{y}\right)}\), I need to eliminate the denominator
• Strategy: Multiply both sides by y to isolate x

3. SIMPLIFY by multiplying both sides by y

• Left side: \(\mathrm{y(b - 72)}\)
• Right side: \(\mathrm{y\left(\frac{x}{y}\right) = x}\)
• Result: \(\mathrm{y(b - 72) = x}\)

4. SIMPLIFY by applying distributive property

• Distribute y to both terms: \(\mathrm{y(b - 72) = y \cdot b + y \cdot (-72)}\)
• This gives us: \(\mathrm{by - 72y = x}\)
• Therefore: \(\mathrm{x = by - 72y}\)

Answer: D. x = by - 72y

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret the equation structure and think that if \(\mathrm{b - 72 = \frac{x}{y}}\), then \(\mathrm{x = \frac{b - 72}{y}}\). They incorrectly assume they should divide (b - 72) by y rather than recognizing that \(\mathrm{\frac{x}{y}}\) equals (b - 72), so x equals y times (b - 72).

This leads them to select Choice A \(\mathrm{\left(\frac{b - 72}{y}\right)}\).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly multiply both sides by y to get \(\mathrm{y(b - 72) = x}\), but then fail to fully distribute. They might write \(\mathrm{y(b - 72) = yb - 72}\), forgetting that the y must multiply the entire expression (b - 72), including the -72 term.

This may lead them to select Choice B \(\mathrm{(by - 72)}\).

The Bottom Line:

This problem tests whether students can systematically work backwards from a fraction to isolate the numerator variable, requiring both strategic thinking about equation structure and careful execution of the distributive property.