The table gives the number of hours, h, of labor and a plumber's total charge \(\mathrm{f(h)}\), in dollars, for two...

1. TRANSLATE the problem information

• Given information:
  • Two points: \((1, 155)\) and \((3, 285)\)
  • Linear relationship between h and f(h)
  • Need to find equation in form \(\mathrm{f(h) = mh + b}\)

2. INFER the solution strategy

• Since we have two points on a linear function, we can:
  • First find the slope (rate of change)
  • Then use either point to find the y-intercept

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{285 - 155}{3 - 1}}\)
• \(\mathrm{m = \frac{130}{2}}\)
• \(\mathrm{m = 65}\)

4. SIMPLIFY to find the y-intercept

• Substitute slope and one point into \(\mathrm{f(h) = mh + b}\)
• Using point \((1, 155)\): \(\mathrm{155 = 65(1) + b}\)
• \(\mathrm{155 = 65 + b}\)
• \(\mathrm{b = 90}\)

5. Write the final equation

• \(\mathrm{f(h) = 65h + 90}\)

Answer: C. \(\mathrm{f(h) = 65h + 90}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students calculate the change in f(h) correctly as \(\mathrm{285 - 155 = 130}\), but forget to divide by the change in h.

They think: "The rate of change is 130" and write \(\mathrm{f(h) = 130h + b}\). Then using point \((1, 155)\): \(\mathrm{155 = 130(1) + b}\), so \(\mathrm{b = 25}\).

This may lead them to select Choice B (\(\mathrm{f(h) = 130h + 25}\))

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating the slope or solving for b, leading to incorrect values that happen to match wrong answer choices.

For example, miscalculating \(\mathrm{130 \div 2}\) or making sign errors when solving \(\mathrm{155 = 65 + b}\).

This may lead them to select Choice A (\(\mathrm{f(h) = 25h + 130}\)) or Choice D (\(\mathrm{f(h) = 90h + 65}\))

The Bottom Line:

This problem tests whether students can systematically apply the two-step process for finding linear equations: calculate slope correctly, then substitute carefully to find the y-intercept. The most dangerous error is treating the numerator of the slope calculation as the final slope value.