Question:A function f is defined for all real numbers x ≠ -3 by \(\mathrm{f(x) = \frac{mx + n}{x + 3}}\),...

1. TRANSLATE the given conditions into equations

• Given information:
  • Function: \(\mathrm{f(x) = \frac{mx + n}{x + 3}}\)
  • \(\mathrm{f(1) = 7}\) means when \(\mathrm{x = 1}\), the function equals 7
  • \(\mathrm{f(5) = 11}\) means when \(\mathrm{x = 5}\), the function equals 11
• What this tells us: We can substitute these x-values into our function to create equations with m and n.

2. TRANSLATE each condition into an algebraic equation

For \(\mathrm{f(1) = 7}\):

• Substitute \(\mathrm{x = 1}\): \(\mathrm{f(1) = \frac{m \cdot 1 + n}{1 + 3} = \frac{m + n}{4} = 7}\)
• This gives us: \(\mathrm{m + n = 28}\)

For \(\mathrm{f(5) = 11}\):

• Substitute \(\mathrm{x = 5}\): \(\mathrm{f(5) = \frac{m \cdot 5 + n}{5 + 3} = \frac{5m + n}{8} = 11}\)
• This gives us: \(\mathrm{5m + n = 88}\)

3. INFER the solution strategy

• We now have a system of two linear equations with two unknowns:
  • \(\mathrm{m + n = 28}\)
  • \(\mathrm{5m + n = 88}\)
• The elimination method works well here since both equations have "+n"

4. SIMPLIFY by solving the system using elimination

• Subtract the first equation from the second:
  \(\mathrm{(5m + n) - (m + n) = 88 - 28}\)
  \(\mathrm{4m = 60}\)
  \(\mathrm{m = 15}\)
• Substitute \(\mathrm{m = 15}\) back into the first equation:
  \(\mathrm{15 + n = 28}\)
  \(\mathrm{n = 13}\)

5. Calculate the final answer

• \(\mathrm{mn = 15 \times 13 = 195}\) (use calculator)

Answer: 195

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert \(\mathrm{f(1) = 7}\) into the equation \(\mathrm{\frac{m + n}{4} = 7}\), often forgetting to substitute \(\mathrm{x = 1}\) into both the numerator and denominator correctly.

They might write something like "\(\mathrm{m + n = 7}\)" directly, missing the division by 4. This leads to an incorrect system of equations and ultimately a wrong value for mn. Since this is a grid-in question, this error leads to confusion and incorrect numerical answers.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the correct system but make arithmetic errors during elimination, particularly when subtracting equations or computing the final product.

For example, they might incorrectly calculate \(\mathrm{4m = 60}\) as \(\mathrm{m = 12}\) instead of \(\mathrm{m = 15}\), leading to wrong values for both m and n, and consequently an incorrect product mn.

The Bottom Line:

This problem tests whether students can systematically translate function conditions into algebraic equations and solve the resulting system. The key insight is recognizing that each given function value creates one linear equation in the two unknowns m and n.