Question:The function p is linear. It satisfies \(\mathrm{p(4) + p(10) = 18}\) and \(\mathrm{p(10) - p(4) = -12}\). Which equation...

1. TRANSLATE the problem information

• Given information:
  • p is a linear function
  • \(\mathrm{p(4) + p(10) = 18}\)
  • \(\mathrm{p(10) - p(4) = -12}\)

• Since p is linear: \(\mathrm{p(x) = mx + b}\) where m is slope and b is y-intercept

2. INFER the most efficient approach

• Key insight: The difference equation \(\mathrm{p(10) - p(4) = -12}\) will eliminate the b term, letting us find m directly
• Strategy: Solve for slope first, then use substitution to find y-intercept

3. TRANSLATE and SIMPLIFY the difference equation

• \(\mathrm{p(10) - p(4) = -12}\) becomes:

\(\mathrm{(10m + b) - (4m + b) = -12}\)

• Expand:

\(\mathrm{10m + b - 4m - b = -12}\)

• SIMPLIFY:

\(\mathrm{6m = -12}\)

• Therefore:

\(\mathrm{m = -2}\)

4. TRANSLATE and SIMPLIFY the sum equation

• \(\mathrm{p(4) + p(10) = 18}\) becomes:

\(\mathrm{(4m + b) + (10m + b) = 18}\)

• SIMPLIFY:

\(\mathrm{14m + 2b = 18}\)

• Substitute \(\mathrm{m = -2}\):

\(\mathrm{14(-2) + 2b = 18}\)

• SIMPLIFY:

\(\mathrm{-28 + 2b = 18}\)

• SIMPLIFY:

\(\mathrm{2b = 46}\), so \(\mathrm{b = 23}\)

5. INFER the final answer

• \(\mathrm{p(x) = mx + b = -2x + 23}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may try to substitute specific x-values randomly instead of setting up the systematic equations \(\mathrm{p(x) = mx + b}\).

For example, they might attempt to guess-and-check using the answer choices rather than TRANSLATE the conditions into: \(\mathrm{(10m + b) - (4m + b) = -12}\) and \(\mathrm{(4m + b) + (10m + b) = 18}\). This leads to inefficient work and potential selection of any incorrect choice through guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors, particularly when expanding \(\mathrm{(10m + b) - (4m + b)}\) or when solving \(\mathrm{-28 + 2b = 18}\).

A common mistake is: \(\mathrm{(10m + b) - (4m + b) = 10m + b - 4m + b = 6m + 2b = -12}\), forgetting to distribute the negative sign. This leads to wrong values for m and b, potentially causing them to select Choice A (\(\mathrm{p(x) = 2x - 5}\)) or another incorrect option.

The Bottom Line:

This problem rewards systematic algebraic thinking over guess-and-check approaches. The key insight is recognizing that linear function problems with two conditions naturally lead to a system of two equations in two unknowns.