The function f is defined by \(\mathrm{f(x) = \frac{2x - 1}{3}}\). For what value of x does \(\mathrm{f(x) = 7}\)?410111222

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = \frac{2x - 1}{3}}\)
  • Need to find: the value of x where \(\mathrm{f(x) = 7}\)

• This means we need: \(\mathrm{\frac{2x - 1}{3} = 7}\)

2. SIMPLIFY through algebraic steps

• Start with: \(\mathrm{\frac{2x - 1}{3} = 7}\)

• Multiply both sides by 3 to eliminate the fraction:
  \(\mathrm{2x - 1 = 21}\)

• Add 1 to both sides to isolate the term with x:
  \(\mathrm{2x = 22}\)

• Divide both sides by 2 to solve for x:
  \(\mathrm{x = 11}\)

3. Verify the answer

• Check: \(\mathrm{f(11) = \frac{2(11) - 1}{3}}\)
  \(\mathrm{= \frac{22 - 1}{3}}\)
  \(\mathrm{= \frac{21}{3}}\)
  \(\mathrm{= 7}\) ✓

Answer: C. 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{\frac{2x - 1}{3} = 7}\) and work through most steps, but forget to complete the final division step.

After reaching \(\mathrm{2x = 22}\), they stop and think \(\mathrm{x = 22}\), missing that they still need to divide both sides by 2. This may lead them to select Choice E (22).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the multi-step algebra, such as incorrectly handling the multiplication by 3 or making sign errors when adding 1 to both sides.

These calculation mistakes can lead to various incorrect x-values, causing them to select one of the other wrong answer choices or get confused and guess.

The Bottom Line:

This problem tests systematic algebraic manipulation more than conceptual understanding. Success requires carefully executing each inverse operation step while maintaining equation balance.