3a = b - 7cThe given equation relates the distinct positive integers a, b, and c. Which equation correctly expresses...

1. TRANSLATE the problem requirement

• Given: \(\mathrm{3a = b - 7c}\)
• Need to find: \(\mathrm{a}\) expressed in terms of \(\mathrm{b}\) and \(\mathrm{c}\)
• This means: isolate \(\mathrm{a}\) on one side of the equation

2. SIMPLIFY by dividing both sides by 3

• To isolate \(\mathrm{a}\), divide both sides by 3:

\(\mathrm{3a ÷ 3 = (b - 7c) ÷ 3}\)

• Left side: \(\mathrm{3a ÷ 3 = a}\)
• Right side: \(\mathrm{(b - 7c) ÷ 3 = \frac{b - 7c}{3}}\)

3. Write the final result

• \(\mathrm{a = \frac{b - 7c}{3}}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly distribute the division by 3, dividing only part of the right side instead of the entire expression.

They might think: "I need to divide by 3, so \(\mathrm{b ÷ 3 = \frac{b}{3}}\) and the rest stays the same"

This incorrect reasoning leads to: \(\mathrm{a = \frac{b}{3} - 7c}\)

This may lead them to select Choice A (\(\mathrm{a = \frac{b}{3} - 7c}\))

Second Most Common Error:

Poor SIMPLIFY execution: Students divide only the constant term by 3, leaving \(\mathrm{b}\) unchanged.

They reason: "The \(\mathrm{7c}\) part needs to be divided by 3" but forget that division applies to the entire expression.

This leads to: \(\mathrm{a = b - \frac{7c}{3}}\)

This may lead them to select Choice B (\(\mathrm{a = b - \frac{7c}{3}}\))

The Bottom Line:

The key challenge is remembering that when dividing both sides of an equation, the division must apply to the entire expression on each side, not just individual terms. Parentheses help maintain this proper grouping.