Question:A factory has scheduled 145 units of production for the day. The equation 12x + 17y = 145 represents this...

1. TRANSLATE the problem information

• Given information:
  • Factory scheduled \(145\) units total production
  • Equation: \(12\mathrm{x} + 17\mathrm{y} = 145\)
  • \(\mathrm{x}\) = number of type A machines scheduled
  • \(\mathrm{y}\) = number of type B machines scheduled
  • Need to find: How many more units each type B machine produces than each type A machine

2. INFER what the coefficients represent

• In the equation \(12\mathrm{x} + 17\mathrm{y} = 145\):
  • The coefficient \(12\) tells us each type A machine produces \(12\) units
  • The coefficient \(17\) tells us each type B machine produces \(17\) units
• To answer "how many more," we need the difference between these production rates

3. Calculate the difference

• Type B production per machine: \(17\) units
• Type A production per machine: \(12\) units
• Difference: \(17 - 12 = 5\) units

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what the question is asking and attempt to solve for specific values of \(\mathrm{x}\) and \(\mathrm{y}\) instead of comparing the coefficients.

They might try finding integer solutions to \(12\mathrm{x} + 17\mathrm{y} = 145\), getting lost in complex algebraic work when the answer is simply comparing the coefficients \(17\) and \(12\). This leads to confusion and abandoning the systematic approach to guess.

Second Most Common Error:

Missing conceptual knowledge about coefficients: Students don't recognize that coefficients represent production rates per machine.

They might interpret \(12\) and \(17\) as total machines or total production rather than units per machine. This conceptual confusion prevents them from identifying what numbers to compare, causing them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests interpretation of coefficients in real-world linear equation contexts, not algebraic equation-solving skills. The key insight is recognizing what the numbers in the equation actually represent.