The equation C = 25m + 60 models the total cost, C, in dollars, for a mobile phone plan that...

1. INFER the mathematical structure

• Given information:
  • Cost equation: \(\mathrm{C = 25m + 60}\)
  • Total amount paid: \(\$410\)
• This creates a system of two equations where we need to find both m and C

2. INFER the solution strategy

• We have two equations with the same variable C:
  • \(\mathrm{C = 25m + 60}\)
  • \(\mathrm{C = 410}\)
• Since both expressions equal C, we can use substitution: set them equal to each other

3. SIMPLIFY through substitution and algebra

• Substitute: \(\mathrm{410 = 25m + 60}\)
• Subtract 60 from both sides: \(\mathrm{410 - 60 = 25m}\)
• Simplify: \(\mathrm{350 = 25m}\)
• Divide by 25: \(\mathrm{m = 350 ÷ 25 = 14}\)

4. INFER the complete solution

• We found \(\mathrm{m = 14}\)
• We know \(\mathrm{C = 410}\) (given)
• The solution as an ordered pair is \(\mathrm{(m, C) = (14, 410)}\)

Answer: B. (14, 410)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a system of equations problem. They might try to solve \(\mathrm{C = 25m + 60}\) for C without using the \(\$410\) information, or they get confused about how to combine the two pieces of information.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the coordinate notation and think the answer should be \(\mathrm{(C, m)}\) instead of \(\mathrm{(m, C)}\). They correctly find \(\mathrm{m = 14}\) and \(\mathrm{C = 410}\), but write the answer as \(\mathrm{(410, 14)}\).

This may lead them to select Choice D (410, 14).

The Bottom Line:

This problem tests whether students can recognize that real-world constraints (the \(\$410\) payment) create additional equations in a system, not just individual values to plug in.