The table shows the linear relationship between the number of cars, c, on a commuter train and the maximum number...

1. TRANSLATE the table information

• Given information: Table showing linear relationship between c (cars) and p (passengers/crew)
  • Point 1: \((3, 174)\)
  • Point 2: \((5, 284)\)
  • Point 3: \((10, 559)\)

• What this tells us: We need to find an equation representing this linear relationship

2. INFER the solution approach

• Since we have a linear relationship, the equation will be in the form \(\mathrm{p = mc + b}\)
• We need to find the slope (m) and y-intercept (b) first
• Then rearrange to match one of the given answer choice formats

3. Calculate the slope

• Using any two points from the table:
• Slope = \(\frac{284 - 174}{5 - 3}\)

\(= \frac{110}{2}\)

\(= 55\)

4. Find the y-intercept

• Substitute slope and any point into \(\mathrm{p = mc + b}\)
• Using point \((3, 174)\):

\(174 = 55(3) + \mathrm{b}\)

\(174 = 165 + \mathrm{b}\)

\(\mathrm{b} = 9\)

5. SIMPLIFY to get the final equation form

• We now have: \(\mathrm{p = 55c + 9}\)
• INFER that we need to rearrange this to match the answer choices
• Subtract 9 from both sides: \(\mathrm{p - 9 = 55c}\)
• Subtract p from both sides: \(-9 = 55\mathrm{c} - \mathrm{p}\)
• Therefore: \(55\mathrm{c} - \mathrm{p} = -9\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly find \(\mathrm{p = 55c + 9}\) but make sign errors during algebraic rearrangement. They might incorrectly rearrange to get \(55\mathrm{c} - \mathrm{p} = 9\) instead of \(55\mathrm{c} - \mathrm{p} = -9\), mixing up which side the constant belongs on.

This may lead them to select Choice B \((55\mathrm{c} - \mathrm{p} = 9)\)

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret which variable should be treated as the independent variable, thinking p should be expressed in terms of c in a different format. They might try to solve for c in terms of p, leading to equations with 55p in them.

This may lead them to select Choice C \((55\mathrm{p} - \mathrm{c} = -9)\) or Choice D \((55\mathrm{p} - \mathrm{c} = 9)\)

The Bottom Line:

The key challenge is managing the multiple algebraic steps needed to rearrange from slope-intercept form to standard form while keeping track of positive and negative signs correctly.