A line in the xy-plane has a slope of -{1/4} and an x-intercept of 12. Which equation represents this line?

1. TRANSLATE the problem information

• Given information:
  • Slope = \(\mathrm{-\frac{1}{4}}\)
  • x-intercept = \(\mathrm{12}\)
• What this tells us: The line passes through the point \(\mathrm{(12, 0)}\) because x-intercept means \(\mathrm{y = 0}\) when \(\mathrm{x = 12}\)

2. INFER the approach

• We need to find the equation of the line in slope-intercept form: \(\mathrm{y = mx + b}\)
• We know \(\mathrm{m = -\frac{1}{4}}\), but we need to find \(\mathrm{b}\) (the y-intercept)
• Since we have a point on the line \(\mathrm{(12, 0)}\) and the slope, we can substitute these values to solve for \(\mathrm{b}\)

3. SIMPLIFY to find the y-intercept

• Substitute the point \(\mathrm{(12, 0)}\) and slope into \(\mathrm{y = mx + b}\):
  \(\mathrm{0 = (-\frac{1}{4})(12) + b}\)
• Calculate:
  \(\mathrm{0 = -3 + b}\)
• Solve for \(\mathrm{b}\):
  \(\mathrm{b = 3}\)

4. Write the final equation

• Now we have \(\mathrm{m = -\frac{1}{4}}\) and \(\mathrm{b = 3}\)
• The equation is \(\mathrm{y = (-\frac{1}{4})x + 3}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse x-intercept with y-intercept and think the line passes through \(\mathrm{(0, 12)}\) instead of \(\mathrm{(12, 0)}\).

When they substitute \(\mathrm{(0, 12)}\) into \(\mathrm{y = mx + b}\):
\(\mathrm{12 = (-\frac{1}{4})(0) + b}\)
\(\mathrm{12 = 0 + b}\)
\(\mathrm{b = 12}\)

This leads them to the equation \(\mathrm{y = (-\frac{1}{4})x + 12}\), causing them to select Choice E.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when calculating \(\mathrm{(-\frac{1}{4})(12)}\).

They might calculate \(\mathrm{(-\frac{1}{4})(12) = +3}\) instead of \(\mathrm{-3}\), leading to:
\(\mathrm{0 = 3 + b}\)
\(\mathrm{b = -3}\)

This gives them \(\mathrm{y = (-\frac{1}{4})x - 3}\), causing them to select Choice A.

The Bottom Line:

The key challenge is correctly interpreting what "x-intercept of 12" means geometrically and then executing the algebra without sign errors. Students who master the TRANSLATE skill of converting intercepts to coordinate points will find this problem straightforward.