QUESTION STEM:A taxi fare is modeled by \(\mathrm{T(d) = ad + c}\), where d is the distance in miles and...

1. TRANSLATE the problem information

• Given information:
  • Taxi fare function: \(\mathrm{T(d) = a \cdot d + c}\)
  • 2-mile trip costs \(\$13\)
  • 5-mile trip costs \(\$22\)
  • Need to find: value of a

2. TRANSLATE the scenarios into equations

• Substitute the given trip data into the function:
  • For 2 miles: \(\mathrm{T(2) = 2a + c = 13}\)
  • For 5 miles: \(\mathrm{T(5) = 5a + c = 22}\)
• This gives us a system of two equations with two unknowns (a and c)

3. INFER the best solution strategy

• We have:
  • \(\mathrm{2a + c = 13}\)
  • \(\mathrm{5a + c = 22}\)
• Since both equations have the same "+c" term, elimination by subtraction will directly solve for a
• This avoids having to find c first

4. SIMPLIFY by eliminating c

• Subtract the first equation from the second:
  \(\mathrm{(5a + c) - (2a + c) = 22 - 13}\)
• The c terms cancel out:
  \(\mathrm{5a + c - 2a - c = 9}\)
  \(\mathrm{3a = 9}\)
• Divide both sides by 3:
  \(\mathrm{a = 3}\)

5. Verify the solution (optional but recommended)

• If \(\mathrm{a = 3}\), then from the first equation: \(\mathrm{c = 13 - 2(3) = 7}\)
• Check with second equation: \(\mathrm{5(3) + 7 = 22}\) ✓

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may try to work with just one equation instead of recognizing they need both pieces of information to create a solvable system.

They might use only \(\mathrm{T(2) = 13}\) and think they can somehow find a directly, not realizing they have two unknowns (a and c) but only one equation. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER strategy: Students correctly set up the system but choose substitution instead of elimination, making the algebra more complex and error-prone.

They might solve \(\mathrm{2a + c = 13}\) for c (getting \(\mathrm{c = 13 - 2a}\)), then substitute into the second equation, leading to more chances for arithmetic mistakes during the SIMPLIFY phase.

The Bottom Line:

This problem tests whether students can efficiently translate real-world linear relationships into mathematical systems and choose the most direct solution path. The key insight is recognizing that elimination immediately isolates the target variable.