c - 7 = 25p + kThe given equation relates the positive numbers c, p, and k. Which equation correctly...

1. TRANSLATE the problem requirement

• Given equation: \(\mathrm{c - 7 = 25p + k}\)
• Goal: Express \(\mathrm{c}\) in terms of \(\mathrm{p}\) and \(\mathrm{k}\) (meaning solve for \(\mathrm{c}\))

2. SIMPLIFY by isolating the variable c

• Currently \(\mathrm{c}\) has \(\mathrm{-7}\) attached to it
• To isolate \(\mathrm{c}\), add 7 to both sides of the equation:

\(\mathrm{c - 7 + 7 = 25p + k + 7}\)

• Simplifying the left side: \(\mathrm{c = 25p + k + 7}\)

Answer: A. \(\mathrm{c = 25p + k + 7}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students get confused about which operation to perform and subtract 7 from both sides instead of adding 7.

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

\(\mathrm{c - 7 - 7 = 25p + k - 7}\)

leading to

\(\mathrm{c - 14 = 25p + k - 7}\)

then

\(\mathrm{c = 25p + k - 7 + 14 = 25p + k + 7}\)

Wait, this actually leads to the right answer through a more complicated path.

Actually, the more direct error is: "I see \(\mathrm{-7}\) on the left, so the answer should have \(\mathrm{-7}\)," without properly working through the algebra.

This may lead them to select Choice B (\(\mathrm{c = 25p + k - 7}\)).

The Bottom Line:

This is a fundamental algebra problem that tests whether students understand the inverse operation concept: to undo subtraction of 7, you must add 7 to both sides. The key insight is recognizing that the \(\mathrm{-7}\) is being subtracted from \(\mathrm{c}\), so adding 7 eliminates it.