Question:The functions a and b are defined by \(\mathrm{a(x) = \frac{3}{8}x - 6}\) and \(\mathrm{b(x) = \frac{2}{8}x - 3}\). For...

1. TRANSLATE the problem requirement

• Given information:
  • \(\mathrm{a(x) = \frac{3}{8}x - 6}\)
  • \(\mathrm{b(x) = \frac{2}{8}x - 3}\)
  • Need to find where functions have "same output value"

• What this means: We need \(\mathrm{a(x) = b(x)}\)

2. TRANSLATE into an equation

• Set the functions equal to each other:
  \(\mathrm{\frac{3}{8}x - 6 = \frac{2}{8}x - 3}\)

3. SIMPLIFY by collecting like terms

• Move all x-terms to the left side by subtracting \(\mathrm{\frac{2}{8}x}\):
  \(\mathrm{\frac{3}{8}x - \frac{2}{8}x - 6 = -3}\)
• Combine the fractions: \(\mathrm{\frac{3}{8} - \frac{2}{8} = \frac{1}{8}}\)
  \(\mathrm{\frac{1}{8}x - 6 = -3}\)

4. SIMPLIFY to isolate the variable

• Add 6 to both sides:
  \(\mathrm{\frac{1}{8}x = -3 + 6 = 3}\)
• Multiply both sides by 8:
  \(\mathrm{x = 3 \times 8 = 24}\)

Answer: 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding that "same output value" means setting the functions equal to each other.

Instead of creating the equation \(\mathrm{a(x) = b(x)}\), students might try to substitute specific values or solve each function separately. Without the key insight that equal outputs means \(\mathrm{a(x) = b(x)}\), they cannot establish the fundamental equation needed to solve the problem. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when combining fractions or moving terms across the equation.

Common mistakes include getting the signs wrong when moving -6 and -3, or incorrectly combining \(\mathrm{\frac{3}{8}x - \frac{2}{8}x}\). For example, they might calculate \(\mathrm{\frac{3}{8} - \frac{2}{8} = \frac{1}{16}}\) instead of \(\mathrm{\frac{1}{8}}\), leading to \(\mathrm{x = 48}\) instead of 24. This causes them to enter an incorrect numerical answer.

The Bottom Line:

This problem tests whether students can translate the concept of "equal function outputs" into a solvable equation, then execute the algebraic steps accurately. The translation step is critical - without it, no progress is possible.