6r = 7s + t The given equation relates the variables r, s, and t. Which equation correctly expresses s...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{6r = 7s + t}\)
• Goal: Express \(\mathrm{s}\) in terms of \(\mathrm{r}\) and \(\mathrm{t}\) (isolate \(\mathrm{s}\))

2. INFER the solution strategy

• Since \(\mathrm{s}\) is multiplied by 7 and has \(\mathrm{t}\) added to it, I need to "undo" these operations
• Strategy: Move \(\mathrm{t}\) away from \(\mathrm{s}\), then divide by the coefficient of \(\mathrm{s}\)

3. SIMPLIFY by removing t from the right side

• Subtract \(\mathrm{t}\) from both sides: \(\mathrm{6r - t = 7s}\)
• Now \(\mathrm{s}\) is isolated on the right side except for the coefficient 7

4. SIMPLIFY by removing the coefficient of s

• Divide both sides by 7: \(\mathrm{\frac{6r - t}{7} = s}\)
• Therefore: \(\mathrm{s = \frac{6r - t}{7}}\)

Answer: D. \(\mathrm{s = \frac{6r - t}{7}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly handle the division by 7 step.

Instead of dividing the entire expression \(\mathrm{(6r - t)}\) by 7, they might only divide one term, leading to expressions like \(\mathrm{s = \frac{6r}{7} - t}\). This reasoning error stems from not understanding that when dividing both sides by 7, the entire left side expression must be divided by 7.

This may lead them to select Choice C (\(\mathrm{s = \frac{6}{7}r - t}\)).

Second Most Common Error:

Poor INFER reasoning about inverse operations: Students multiply by 7 instead of dividing by 7.

They correctly move \(\mathrm{t}\) to get \(\mathrm{6r - t = 7s}\), but then think "to get rid of the 7, I multiply both sides by 7" rather than understanding that division undoes multiplication. This leads to \(\mathrm{s = 7(6r - t)}\).

This may lead them to select Choice B (\(\mathrm{s = 7(6r - t)}\)).

The Bottom Line:

This problem tests whether students can systematically apply inverse operations while maintaining algebraic precision. The key insight is that each step must be applied to the entire expression, not just individual terms.