The function f is defined by \(\mathrm{f(x) = \frac{9}{7}x + \frac{8}{7}}\). For what value of x does \(\mathrm{f(x) = 5}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = \frac{9}{7}x + \frac{8}{7}}\)
  • Condition: \(\mathrm{f(x) = 5}\)
  • Need to find: value of x

• This tells us we need to set the function equal to 5 and solve for x

2. TRANSLATE to create the equation

• Since \(\mathrm{f(x) = 5}\), we substitute:
  \(\mathrm{5 = \frac{9}{7}x + \frac{8}{7}}\)

3. SIMPLIFY by clearing fractions

• Multiply both sides by 7 to eliminate denominators:
  \(\mathrm{7(5) = 7(\frac{9}{7}x + \frac{8}{7})}\)
  \(\mathrm{35 = 9x + 8}\)

• This makes the equation much easier to work with

4. SIMPLIFY to isolate the x term

• Subtract 8 from both sides:
  \(\mathrm{35 - 8 = 9x}\)
  \(\mathrm{27 = 9x}\)

5. SIMPLIFY to find x

• Divide both sides by 9:
  \(\mathrm{x = \frac{27}{9} = 3}\)

Answer: 3 (also acceptable: 3.0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when multiplying by 7 or performing subsequent calculations. For example, they might incorrectly calculate \(\mathrm{7 × 5 = 30}\) instead of \(\mathrm{35}\), or make mistakes when subtracting 8 or dividing by 9. These calculation errors lead to wrong values like \(\mathrm{x = 2}\) or \(\mathrm{x = 4}\), causing confusion and incorrect answer selection.

Second Most Common Error:

Poor TRANSLATE reasoning: Students struggle to properly set up the initial equation. They might confuse which expression equals 5, potentially writing \(\mathrm{x = \frac{9}{7}(5) + \frac{8}{7}}\) instead of \(\mathrm{5 = \frac{9}{7}x + \frac{8}{7}}\). This fundamental setup error leads them down the wrong solution path entirely, causing them to get stuck and guess.

The Bottom Line:

This problem requires careful attention to equation setup and systematic algebraic manipulation. The fraction coefficients make arithmetic more prone to errors, but the underlying linear equation solving process remains straightforward once properly established.