For the linear function \(\mathrm{j(x) = mx + 144}\), m is a constant and \(\mathrm{j(12) = 18}\). What is the...

1. TRANSLATE the given information

• Given information:
  • Linear function: \(\mathrm{j(x) = mx + 144}\)
  • \(\mathrm{m}\) is a constant (unknown value)
  • \(\mathrm{j(12) = 18}\) (when \(\mathrm{x = 12}\), the function equals 18)
  • Need to find: \(\mathrm{j(10)}\)

• What this tells us: We can use \(\mathrm{j(12) = 18}\) to find the value of \(\mathrm{m}\) first

2. INFER the solution strategy

• Since we have one unknown (\(\mathrm{m}\)) and one condition (\(\mathrm{j(12) = 18}\)), we can:
  • Substitute the known values into the function to create an equation
  • Solve for \(\mathrm{m}\)
  • Use the complete function to find \(\mathrm{j(10)}\)

3. TRANSLATE the condition into an equation

• Substitute \(\mathrm{x = 12}\) and \(\mathrm{j(x) = 18}\) into \(\mathrm{j(x) = mx + 144}\):

\(\mathrm{18 = m(12) + 144}\)

4. SIMPLIFY to solve for m

• Start with: \(\mathrm{18 = 12m + 144}\)
• Subtract 144 from both sides: \(\mathrm{18 - 144 = 12m}\)
• Calculate: \(\mathrm{-126 = 12m}\)
• Divide by 12: \(\mathrm{m = -126/12 = -10.5}\)

5. TRANSLATE the complete function

• Now we know: \(\mathrm{j(x) = -10.5x + 144}\)

6. SIMPLIFY to find j(10)

• Substitute \(\mathrm{x = 10}\): \(\mathrm{j(10) = -10.5(10) + 144}\)
• Calculate: \(\mathrm{j(10) = -105 + 144 = 39}\)

Answer: 39

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might try to find \(\mathrm{j(10)}\) directly without first determining the value of \(\mathrm{m}\). They see \(\mathrm{j(x) = mx + 144}\) and \(\mathrm{j(10)}\), so they might write \(\mathrm{j(10) = m(10) + 144 = 10m + 144}\), but then get stuck because they don't know what \(\mathrm{m}\) equals. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{18 = 12m + 144}\) but make arithmetic errors when solving for \(\mathrm{m}\). Common mistakes include:

• Getting the wrong sign when subtracting: \(\mathrm{18 + 144 = 162}\) instead of \(\mathrm{18 - 144 = -126}\)
• Division errors: calculating \(\mathrm{-126 \div 12}\) incorrectly

These computational errors lead to wrong values of \(\mathrm{m}\) and consequently wrong final answers.

The Bottom Line:

This problem requires recognizing that you must use the given condition to find the missing parameter before you can evaluate the function at a different point. Students who jump ahead without this crucial first step will find themselves unable to proceed systematically.