q - 29r = s The given equation relates the positive numbers q, r, and s. Which equation correctly expresses...

1. TRANSLATE the problem requirement

• Given: \(\mathrm{q - 29r = s}\)
• Find: Express \(\mathrm{q}\) in terms of \(\mathrm{r}\) and \(\mathrm{s}\)
• This means: Isolate \(\mathrm{q}\) on one side of the equation

2. SIMPLIFY by isolating q

• Currently: \(\mathrm{q - 29r = s}\)
• To isolate \(\mathrm{q}\), I need to "undo" the subtraction of \(\mathrm{29r}\)
• Add \(\mathrm{29r}\) to both sides: \(\mathrm{q - 29r + 29r = s + 29r}\)
• Left side simplifies: \(\mathrm{q - 29r + 29r = q}\)
• Result: \(\mathrm{q = s + 29r}\)

Answer: B. q = s + 29r

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make a sign error when moving terms across the equals sign.

Instead of adding \(\mathrm{29r}\) to both sides, they might subtract \(\mathrm{29r}\) from both sides:

\(\mathrm{q - 29r - 29r = s - 29r}\)
\(\mathrm{q - 58r = s - 29r}\)

Or they might incorrectly think that moving \(\mathrm{29r}\) to the right side means it stays negative:

\(\mathrm{q = s - 29r}\)

This may lead them to select Choice A (\(\mathrm{q = s - 29r}\)).

Second Most Common Error:

Conceptual confusion about algebraic structure: Students might misinterpret the relationship between the variables entirely.

They might think that since we have \(\mathrm{q}\), \(\mathrm{r}\), and \(\mathrm{s}\) in the original equation, the answer should involve multiplication or division of these terms, leading to expressions like \(\mathrm{q = 29rs}\) or similar combinations.

This may lead them to select Choice C (\(\mathrm{q = 29rs}\)) or get confused and guess.

The Bottom Line:

This problem tests fundamental algebra skills that form the foundation for more complex equations. The key insight is recognizing that solving for a variable means performing inverse operations - if something is subtracted from the variable, you add it to both sides to isolate the variable.