The cost, in dollars, of producing x items is given by the equation c = 5x + 15.What is the...

1. TRANSLATE the problem information

• Given information:
  • Cost equation: \(\mathrm{c = 5x + 15}\)
  • We know that \(\mathrm{c = 35}\)
• What this tells us: We need to substitute 35 for c and solve for x

2. TRANSLATE the substitution step

• Replace c with 35 in the equation:
  \(\mathrm{35 = 5x + 15}\)

3. SIMPLIFY to isolate the variable

• Subtract 15 from both sides:
  \(\mathrm{35 - 15 = 5x + 15 - 15}\)
  \(\mathrm{20 = 5x}\)

• Divide both sides by 5:
  \(\mathrm{20 ÷ 5 = 5x ÷ 5}\)
  \(\mathrm{4 = x}\)

4. TRANSLATE back to ordered pair format

• The solution as an ordered pair is \(\mathrm{(x, c) = (4, 35)}\)

Answer: A. (4, 35)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the algebraic steps, such as incorrectly calculating \(\mathrm{20 ÷ 5}\) or making sign errors when subtracting 15.

For example, they might think \(\mathrm{20 ÷ 5 = 7}\) somehow, leading them to \(\mathrm{x = 7}\).

This may lead them to select Choice B (7, 35).

Second Most Common Error:

Poor TRANSLATE reasoning about ordered pairs: Students solve correctly to get \(\mathrm{x = 4}\) but then get confused about the ordered pair format, thinking the larger number should come first or mixing up which variable represents which position.

They might write \(\mathrm{(35, 4)}\) instead of \(\mathrm{(4, 35)}\).

This may lead them to select Choice D (35, 15) or causes confusion about the format.

The Bottom Line:

This problem tests whether students can systematically substitute a known value and perform algebraic manipulations accurately. The algebraic steps are straightforward, but small arithmetic errors or confusion about ordered pair notation can easily derail the solution.