The linear function p is defined by \(\mathrm{p(x) = 4x - d}\), where d is a constant. If \(\mathrm{p(r +...

1. TRANSLATE the problem setup

• Given information:
  • \(\mathrm{p(x) = 4x - d}\) (linear function)
  • \(\mathrm{p(r + 5) = \frac{r}{3} + 2}\) (condition we must use)
  • Need to find expression for d

2. INFER the solution strategy

• We need to evaluate \(\mathrm{p(r + 5)}\) using our function definition
• Then set it equal to the given condition to solve for d
• This will give us an equation we can solve algebraically

3. SIMPLIFY by substituting into the function

• Substitute (r + 5) into \(\mathrm{p(x) = 4x - d}\):

\(\mathrm{p(r + 5) = 4(r + 5) - d = 4r + 20 - d}\)

4. TRANSLATE the condition into an equation

• We know \(\mathrm{p(r + 5) = \frac{r}{3} + 2}\), so:

\(\mathrm{4r + 20 - d = \frac{r}{3} + 2}\)

5. SIMPLIFY to solve for d

• Rearrange to isolate d:

\(\mathrm{-d = \frac{r}{3} + 2 - 4r - 20}\)

\(\mathrm{-d = \frac{r}{3} - 4r - 18}\)

\(\mathrm{d = 4r - \frac{r}{3} + 18}\)

6. SIMPLIFY by finding common denominators

• Convert 4r to thirds: \(\mathrm{4r = \frac{12r}{3}}\)
• Combine: \(\mathrm{d = \frac{12r}{3} - \frac{r}{3} + 18 = \frac{11r}{3} + 18}\)

Answer: D (\(\mathrm{\frac{11r}{3} + 18}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not properly understand what \(\mathrm{p(r + 5)}\) means, confusing function evaluation with simple substitution. They might try to substitute r + 5 directly for d instead of for x, or they might not recognize that they need to use both pieces of given information together.

This leads to confusion and guessing between the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when rearranging the equation, particularly when moving terms across the equals sign or when distributing the negative sign to isolate d. They might get \(\mathrm{d = -4r + \frac{r}{3} - 18}\) instead of \(\mathrm{d = 4r - \frac{r}{3} + 18}\).

This may lead them to select Choice B (\(\mathrm{\frac{11r}{3} + 22}\)) after incorrectly handling the constant term.

The Bottom Line:

This problem requires understanding that function notation means substitution, and then careful algebraic manipulation. The key insight is recognizing that you have two expressions for the same quantity \(\mathrm{p(r + 5)}\), which allows you to set up an equation to solve for the unknown d.