A cell phone company charges customers based on their monthly data usage. The table shows the data usage d (in...

1. TRANSLATE the table information into coordinate points

• Given information:
  • When data usage \(\mathrm{d = 2}\) GB, monthly charge \(\mathrm{C(d) = \$45}\)
  • When data usage \(\mathrm{d = 5}\) GB, monthly charge \(\mathrm{C(d) = \$75}\)
  • These give us points: \(\mathrm{(2, 45)}\) and \(\mathrm{(5, 75)}\)

2. INFER the solution strategy

• Since we need a linear equation \(\mathrm{C(d) = md + b}\), we must find two unknowns: slope (m) and y-intercept (b)
• Strategy: Calculate slope using the two points, then use point-slope form to find the complete equation

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{75 - 45}{5 - 2}}\)
  \(\mathrm{= \frac{30}{3}}\)
  \(\mathrm{= 10}\)

4. SIMPLIFY using point-slope form to find the equation

• Using point \(\mathrm{(2, 45)}\) and slope \(\mathrm{m = 10}\):
• \(\mathrm{C(d) - 45 = 10(d - 2)}\)
• \(\mathrm{C(d) - 45 = 10d - 20}\)
• \(\mathrm{C(d) = 10d - 20 + 45}\)
• \(\mathrm{C(d) = 10d + 25}\)

5. APPLY CONSTRAINTS by verifying with both data points

• Check: \(\mathrm{C(2) = 10(2) + 25 = 45}\) ✓
• Check: \(\mathrm{C(5) = 10(5) + 25 = 75}\) ✓

Answer: B) \(\mathrm{C(d) = 10d + 25}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic errors when converting from point-slope form to slope-intercept form.

For example, they might incorrectly distribute: \(\mathrm{C(d) - 45 = 10(d - 2)}\) becomes \(\mathrm{C(d) - 45 = 10d - 2}\) instead of \(\mathrm{C(d) - 45 = 10d - 20}\). This leads to \(\mathrm{C(d) = 10d + 43}\), which doesn't match any answer choice and causes confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which variable represents which quantity, treating data usage as the dependent variable instead of the independent variable.

This conceptual confusion leads them to calculate slope incorrectly or attempt to solve for the wrong relationship. When they can't make their work match the given answer choices, they often guess randomly.

The Bottom Line:

This problem tests whether students can systematically work with linear functions using real-world data. Success requires careful translation of table data into mathematical coordinates, followed by precise algebraic manipulation.