When line n is graphed in the xy-plane, it has an x-intercept of \((-4, 0)\) and a y-intercept of \((0,...

1. TRANSLATE the problem information

• Given information:
  • x-intercept: \((-4, 0)\)
  • y-intercept: \((0, 86/3)\)

• What this tells us: We have two specific points that lie on line n

2. INFER the approach

• To find the slope of a line, we need two points and the slope formula
• Since we have two points from the intercepts, we can apply: \(\mathrm{m} = \frac{\mathrm{y_2 - y_1}}{\mathrm{x_2 - x_1}}\)
• Strategy: Use the intercept coordinates as our two points

3. SIMPLIFY using the slope formula

• Set up the formula with our points:
  • Point 1: \((-4, 0)\) → \((\mathrm{x_1}, \mathrm{y_1})\)
  • Point 2: \((0, 86/3)\) → \((\mathrm{x_2}, \mathrm{y_2})\)

• Substitute into slope formula:
  \(\mathrm{m} = \frac{86/3 - 0}{0 - (-4)}\)
  \(\mathrm{m} = \frac{86/3}{4}\)

• SIMPLIFY the fraction:
  \(\mathrm{m} = \frac{86}{3} \times \frac{1}{4} = \frac{86}{12} = \frac{43}{6}\)

Answer: C. 43/6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when dividing the fraction \((86/3)\) by 4.

Instead of computing \(\frac{86}{3} \div 4 = \frac{86}{3} \times \frac{1}{4}\), they might incorrectly multiply: \(\frac{86}{3} \times 4 = \frac{344}{3}\). This computational mistake occurs because students sometimes confuse whether to multiply or divide when working with fractions in the denominator.

This may lead them to select Choice D (344/3)

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up the slope formula but make errors in the final simplification steps.

They might compute \((86/3) \div 4\) correctly as \(86/12\), but then flip the fraction during simplification, getting \(\frac{12}{86} = \frac{6}{43}\) instead of \(\frac{43}{6}\). This happens when students mechanically apply "flip and multiply" rules without careful attention to the order of operations.

This may lead them to select Choice B (6/43)

The Bottom Line:

This problem tests whether students can connect the concept of intercepts to the practical application of the slope formula, then execute the fraction arithmetic correctly. The multiple computational steps create opportunities for simple but costly arithmetic errors.