A machine's value decreases linearly over time. The table shows the machine's value \(\mathrm{V(t)}\), in thousands of dollars, at time...

1. TRANSLATE the problem information

• Given information:
  • Table showing machine value V(t) in thousands of dollars at time t years
  • Values: \(\mathrm{(1, 47), (3, 41), (5, 35)}\)
  • Need to find equation in form \(\mathrm{V(t) = mt + b}\)

• What this tells us: We have three coordinate points from a linear relationship

2. INFER the solution strategy

• Since we need a linear equation \(\mathrm{V(t) = mt + b}\), we must find:
  • Slope (m) using any two points
  • Y-intercept (b) using point-slope form
• Strategy: Calculate slope first, then use point-slope form with one known point

3. SIMPLIFY to find the slope

• Using points \(\mathrm{(1, 47)}\) and \(\mathrm{(3, 41)}\):
• Slope = \(\mathrm{\frac{y_2 - y_1}{x_2 - x_1}}\) = \(\mathrm{\frac{41 - 47}{3 - 1}}\) = \(\mathrm{\frac{-6}{2}}\) = \(\mathrm{-3}\)

4. SIMPLIFY using point-slope form

• Using point \(\mathrm{(1, 47)}\) and slope \(\mathrm{m = -3}\):
• \(\mathrm{V(t) - 47 = -3(t - 1)}\)
• \(\mathrm{V(t) - 47 = -3t + 3}\)
• \(\mathrm{V(t) = -3t + 3 + 47}\)
• \(\mathrm{V(t) = -3t + 50}\)

5. Verify with remaining point

• Check \(\mathrm{(5, 35)}\): \(\mathrm{V(5) = -3(5) + 50 = 35}\) ✓

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly calculate slope as \(\mathrm{\frac{-6}{2}}\) but fail to simplify to \(\mathrm{-3}\), keeping the slope as \(\mathrm{-6}\) instead.

When they use point-slope form with \(\mathrm{m = -6}\) and point \(\mathrm{(1, 47)}\):
\(\mathrm{V(t) - 47 = -6(t - 1)}\) leads to \(\mathrm{V(t) = -6t + 53}\)

This may lead them to select Choice A \(\mathrm{(-6t + 53)}\)

Second Most Common Error:

Poor SIMPLIFY execution: Students calculate the slope correctly as \(\mathrm{-3}\), but make algebraic errors when converting from point-slope form, particularly forgetting to add the constant terms correctly.

For example, they might get \(\mathrm{V(t) = -3t + 47}\) by not properly distributing or combining terms in the point-slope form.

This may lead them to select Choice B \(\mathrm{(-3t + 47)}\)

The Bottom Line:

This problem tests whether students can systematically work through the two-step process of finding a linear equation: calculating slope accurately and then correctly manipulating the point-slope form. The arithmetic and algebraic precision required in both steps creates multiple opportunities for execution errors.