\(7\mathrm{m} = 2(\mathrm{n} + \mathrm{p})\)The given equation relates the positive numbers m, n, and p. Which equation correctly gives m...

1. INFER the solution strategy

• Given: \(\mathrm{7m = 2(n + p)}\)
• Goal: Find \(\mathrm{m}\) in terms of \(\mathrm{n}\) and \(\mathrm{p}\)
• Key insight: \(\mathrm{m}\) is multiplied by \(\mathrm{7}\), so I need to "undo" this multiplication by dividing both sides by \(\mathrm{7}\)

2. SIMPLIFY by applying division property of equality

• Divide both sides by \(\mathrm{7}\):

\(\mathrm{7m ÷ 7 = 2(n + p) ÷ 7}\)

• This gives us:

\(\mathrm{m = \frac{2(n + p)}{7}}\)

Answer: A. \(\mathrm{m = \frac{2(n+p)}{7}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize they need to eliminate the \(\mathrm{7}\), but choose subtraction instead of division.

They think: "I need to get rid of the \(\mathrm{7}\), so I'll subtract \(\mathrm{7}\) from both sides."

This leads to: \(\mathrm{7m - 7 = 2(n + p) - 7}\), which gives them something like \(\mathrm{m = 2(n + p) - 7}\) when they try to continue.

This may lead them to select Choice C (\(\mathrm{m = 2(n + p) - 7}\)).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify that they need to divide by \(\mathrm{7}\), but forget to apply this operation to the entire right side of the equation.

They might write: \(\mathrm{m = 2(n + p)}\) instead of \(\mathrm{m = \frac{2(n + p)}{7}}\), essentially "forgetting" the division step in their final answer.

This may lead them to select Choice B (\(\mathrm{m = 2(n + p)}\)).

The Bottom Line:

This problem tests whether students can correctly identify and execute the inverse operation needed to isolate a variable. The key is recognizing that multiplication by \(\mathrm{7}\) requires division by \(\mathrm{7}\) to undo it, and that this division must be applied to the entire expression on the right side.