The linear function g is defined by \(\mathrm{g(x) = b - 15x}\), where b is a constant. If \(\mathrm{g(c +...

1. TRANSLATE the given condition into a workable equation

• Given information:
  • \(\mathrm{g(x) = b - 15x}\) (linear function with unknown parameter b)
  • \(\mathrm{g(c + 7) = \frac{c}{4}}\) (condition that must be satisfied)
• What this tells us: We need to substitute \(\mathrm{x = c + 7}\) into our function and set it equal to \(\mathrm{\frac{c}{4}}\)

2. TRANSLATE the substitution into an equation

• Substitute \(\mathrm{x = c + 7}\) into \(\mathrm{g(x) = b - 15x}\):
  \(\mathrm{g(c + 7) = b - 15(c + 7)}\)
• Since \(\mathrm{g(c + 7) = \frac{c}{4}}\), we have:
  \(\mathrm{\frac{c}{4} = b - 15(c + 7)}\)

3. SIMPLIFY by applying the distributive property

• Distribute the -15:
  \(\mathrm{\frac{c}{4} = b - 15c - 105}\)

4. SIMPLIFY by isolating the b term

• Add 15c to both sides:
  \(\mathrm{\frac{c}{4} + 15c = b - 105}\)
• Add 105 to both sides:
  \(\mathrm{\frac{c}{4} + 15c + 105 = b}\)

5. SIMPLIFY by combining the c terms

• Convert 15c to fourths: \(\mathrm{15c = \frac{60c}{4}}\)
• Combine: \(\mathrm{\frac{c}{4} + \frac{60c}{4} = \frac{61c}{4}}\)
• Therefore: \(\mathrm{b = \frac{61c}{4} + 105}\)

Answer: C. \(\mathrm{\frac{61c}{4} + 105}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when combining \(\mathrm{\frac{c}{4} + 15c}\), often forgetting to convert 15c to the same denominator or miscalculating \(\mathrm{15 \times 4 = 60}\).

Common mistakes include getting \(\mathrm{\frac{16c}{4}}\) (adding 15 + 1 instead of 60 + 1) or \(\mathrm{\frac{19c}{4}}\) (adding 15 + 4 instead of multiplying). When they get \(\mathrm{\frac{19c}{4} + 105}\), there's no matching choice, but if they drop the 105 term due to confusion, this may lead them to select Choice B (\(\mathrm{\frac{19c}{4} + 7}\)).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the equation but make sign errors or forget steps when isolating b. They might forget to add the 105 back or make errors in the distributive property step.

This leads to partial expressions that don't match the systematic algebraic approach, causing them to get confused and guess among the remaining choices.

The Bottom Line:

This problem requires careful fraction arithmetic combined with systematic algebraic manipulation. Students who rush through the combining of fractions or don't maintain organization in their algebraic steps are most likely to select incorrect answers.