A delivery service models the relationship between distance x (in miles) and total charge y (in dollars) by the linear...

1. TRANSLATE the problem information

• Given information:
  • Linear equation: \(\mathrm{4(y - 1) + 5x = 20}\)
  • \(\mathrm{x}\) represents distance in miles
  • \(\mathrm{y}\) represents total charge in dollars
  • Base charge = total charge when no distance is traveled

• What this tells us: We need to find \(\mathrm{y}\) when \(\mathrm{x = 0}\)

2. SIMPLIFY by substituting the known value

• Since base charge occurs when \(\mathrm{x = 0}\), substitute this into the equation:
  \(\mathrm{4(y - 1) + 5(0) = 20}\)

• This simplifies to:
  \(\mathrm{4(y - 1) = 20}\)

3. SIMPLIFY to solve for y

• Divide both sides by 4:
  \(\mathrm{y - 1 = 5}\)

• Add 1 to both sides:
  \(\mathrm{y = 6}\)

Answer: 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "base charge when no distance is traveled" means setting \(\mathrm{x = 0}\). They might try to solve for \(\mathrm{x}\) instead of \(\mathrm{y}\), or attempt to rearrange the equation into slope-intercept form unnecessarily.

This leads to confusion and potentially guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the algebraic steps, such as incorrectly dividing 20 by 4 (getting 4 instead of 5) or forgetting to add 1 at the final step.

These calculation errors would lead to incorrect answers like 4 or 5.

The Bottom Line:

This problem tests whether students can connect real-world language to mathematical conditions and then execute basic algebraic manipulation accurately. The key insight is recognizing that "no distance traveled" translates to \(\mathrm{x = 0}\).