The table shows four values of x and their corresponding values of \(\mathrm{f(x)}\). There is a linear relationship between x...

1. TRANSLATE the problem information

• Given information:
  • Linear equation form: \(\mathrm{f(x) = mx - 28}\)
  • Table of coordinate pairs showing x and f(x) values
  • Need to find the constant m

• What this tells us: We can substitute any coordinate pair from the table into the equation to create a solvable equation for m

2. TRANSLATE coordinate data into equation

• Choose any coordinate pair from the table (they should all work)
• Using \(\mathrm{(10, 82)}\): When \(\mathrm{x = 10}\), \(\mathrm{f(x) = 82}\)
• Substitute into \(\mathrm{f(x) = mx - 28}\):
  \(\mathrm{82 = m(10) - 28}\)

3. SIMPLIFY to solve for m

• Start with: \(\mathrm{82 = 10m - 28}\)
• Add 28 to both sides: \(\mathrm{82 + 28 = 10m}\)
• Calculate: \(\mathrm{110 = 10m}\)
• Divide by 10: \(\mathrm{m = 11}\)

4. Verify your answer (optional but recommended)

• Test with another point like \(\mathrm{(15, 137)}\):
• \(\mathrm{f(15) = 11(15) - 28}\)
  \(\mathrm{= 165 - 28}\)
  \(\mathrm{= 137}\) ✓

Answer: 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not understand how to connect the table data to the equation form. They might think they need to use all four points simultaneously or try to find a pattern in the table without using the given equation structure.

This leads to confusion and guessing rather than systematic substitution.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors in the algebraic steps, such as:

• Forgetting to add 28 to both sides correctly
• Making calculation errors like 82 + 28 = 100 instead of 110
• Division errors when solving 110 = 10m

This may lead them to select incorrect numerical answers or abandon the solution process.

The Bottom Line:

The key insight is recognizing that each coordinate pair gives you a complete equation to solve - you don't need complex analysis, just careful substitution and algebra.