A ride-hailing service charges a fixed booking fee and a constant rate per mile. The table shows the number of...

1. TRANSLATE the problem information

• Given information:
  • Table with (miles, total cost) pairs: \(\mathrm{(3, 12)}\), \(\mathrm{(6, 19)}\), \(\mathrm{(9, 26)}\)
  • Linear relationship between miles (m) and cost (C)
  • Need equation in the form given in answer choices

2. INFER the approach

• Since this is linear with fixed fee + rate per mile, use form \(\mathrm{C = am + b}\)
• Calculate slope first, then find y-intercept
• Convert final equation to match answer choice format (standard form)

3. SIMPLIFY to find the slope

• Use any two points from the table, such as \(\mathrm{(3, 12)}\) and \(\mathrm{(6, 19)}\)
• Slope = \(\mathrm{\frac{19 - 12}{6 - 3} = \frac{7}{3}}\)

4. SIMPLIFY to find the complete equation

• Start with \(\mathrm{C = \frac{7}{3}m + b}\)
• Use point \(\mathrm{(3, 12)}\): \(\mathrm{12 = \frac{7}{3}(3) + b}\)
• \(\mathrm{12 = 7 + b}\), so \(\mathrm{b = 5}\)
• Therefore: \(\mathrm{C = \frac{7}{3}m + 5}\)

5. SIMPLIFY to convert to standard form

• Multiply entire equation by 3: \(\mathrm{3C = 7m + 15}\)
• Rearrange: \(\mathrm{7m - 3C = -15}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may calculate the slope correctly but fail to recognize they need to convert their slope-intercept form to standard form to match the answer choices. They might stop at \(\mathrm{C = \frac{7}{3}m + 5}\) and not know how to proceed, leading to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might mix up which variable represents which quantity or incorrectly read coordinates from the table. For example, reading \(\mathrm{(12, 3)}\) instead of \(\mathrm{(3, 12)}\) would give slope = \(\mathrm{\frac{19-3}{19-12} = \frac{16}{7}}\), leading to a completely different equation. This may lead them to select Choice B (\(\mathrm{7m - 3C = 15}\)) after making sign errors in their incorrect calculation.

The Bottom Line:

This problem tests whether students can work backwards from data to equation form. The key challenge is recognizing that finding the linear equation is just the first step - you must then manipulate it algebraically to match the given answer format.