A company's monthly internet bill is determined by a flat fee plus a charge for each gigabyte (GB) of data...

1. TRANSLATE the given equation

• Given: \(\mathrm{C - 3g = 75}\)
• \(\mathrm{C}\) = total monthly bill (dollars)
• \(\mathrm{g}\) = gigabytes used over the limit
• We need to interpret what 75 means

2. INFER the best approach to interpret the constant

• To understand what 75 represents, we should rearrange the equation to see it more clearly
• Converting to slope-intercept form (\(\mathrm{C = mg + b}\)) will help us identify each component's meaning

3. SIMPLIFY by rearranging the equation

• Start with: \(\mathrm{C - 3g = 75}\)
• Add 3g to both sides: \(\mathrm{C = 3g + 75}\)

4. INFER the meaning of each part

• Now we have \(\mathrm{C = 3g + 75}\)
• The coefficient \(\mathrm{3}\) means $3 per gigabyte over the limit
• The constant 75 represents the bill when \(\mathrm{g = 0}\) (no extra data used)
• When no data is used over the limit, the bill is $75 - this is the flat fee

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that rearranging the equation helps interpret the constant term.

They might look at \(\mathrm{C - 3g = 75}\) and think "75 is just some number in the equation" without understanding that it represents a specific real-world value. They may randomly guess among the choices or incorrectly think 75 refers to the total bill in all cases.

This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about linear equations: Students might understand that 75 is important but misinterpret what it represents.

They may think that since 75 appears as a standalone number, it must be the total bill (Choice C) rather than recognizing it as the base cost. They don't connect the constant term to the y-intercept concept where it represents the value when the variable equals zero.

This may lead them to select Choice C ($75 total bill).

The Bottom Line:

This problem requires students to move beyond just manipulating the equation and actually interpret what the mathematical components mean in a real-world context. The key insight is recognizing that constants in linear equations represent starting values or fixed costs.