A delivery service models the relationship between the number of packages q and the total cost C (in dollars) with...

1. TRANSLATE the problem information

• Given: Linear equation \(\frac{7\mathrm{q}}{9} + \frac{5\mathrm{C}}{12} = 18\)
• Find: Value of C at the C-intercept

• The C-intercept is where the line crosses the C-axis, which happens when \(\mathrm{q} = 0\)

2. SIMPLIFY by substituting q = 0

• Substitute \(\mathrm{q} = 0\) into the equation:
  \(\frac{7(0)}{9} + \frac{5\mathrm{C}}{12} = 18\)
• This simplifies to: \(\frac{5\mathrm{C}}{12} = 18\)

3. SIMPLIFY to solve for C

• Multiply both sides by 12: \(5\mathrm{C} = 216\)
• Divide both sides by 5: \(\mathrm{C} = \frac{216}{5}\)

Answer: \(\frac{216}{5}\)

Alternative acceptable forms: \(43.2\) or \(43\frac{1}{5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't understand what "C-intercept" means mathematically

Many students see "C-intercept" and either guess randomly or try to solve the equation without setting \(\mathrm{q} = 0\). They might attempt to solve for both variables simultaneously or get confused about which axis corresponds to which intercept.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors in the algebraic steps

Students correctly set \(\mathrm{q} = 0\) but then make calculation mistakes, particularly when multiplying \(\frac{5\mathrm{C}}{12} = 18\) by 12, getting \(5\mathrm{C} = 144\) instead of \(5\mathrm{C} = 216\), leading to \(\mathrm{C} = \frac{144}{5}\) instead of the correct answer.

This may lead them to an incorrect numerical answer.

The Bottom Line:

This problem tests whether students understand coordinate geometry vocabulary (what intercepts mean) and can execute clean algebraic manipulation. The key insight is recognizing that "intercept" has a specific mathematical meaning that translates to setting one variable equal to zero.