In the xy-plane, the points \((-2, 3)\) and \((4, -5)\) lie on the graph of which of the following linear...

1. TRANSLATE the problem information

• Given information:
  • Two points: (-2, 3) and (4, -5)
  • Need to find which linear function contains both points
• What this tells us: Any line passing through both points will have the same slope

2. INFER the approach

• Since all answer choices are in the form f(x) = mx + b, I can find the slope of the line through the given points
• The correct answer will be the choice with this exact slope
• This is more efficient than testing each point in all four equations

3. TRANSLATE the coordinates into the slope formula

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substitute: \(\mathrm{m = \frac{-5 - 3}{4 - (-2)}}\)
• Note: Be careful with the subtraction: \(\mathrm{4 - (-2) = 4 + 2 = 6}\)

4. SIMPLIFY the slope calculation

• \(\mathrm{m = \frac{-8}{6}}\)
• Reduce to lowest terms: \(\mathrm{m = \frac{-4}{3}}\)

5. INFER which answer choice matches

• Check each slope:
  • A: slope = 1
  • B: slope = 1/2
  • C: slope = \(\mathrm{\frac{-4}{3}}\) ✓
  • D: slope = \(\mathrm{\frac{-3}{2}}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students calculate the slope correctly as \(\mathrm{\frac{-8}{6}}\) but fail to reduce it to \(\mathrm{\frac{-4}{3}}\). They then don't recognize that \(\mathrm{\frac{-8}{6}}\) equals \(\mathrm{\frac{-4}{3}}\), so they think none of the answers work.

This leads to confusion and guessing rather than systematic elimination.

Second Most Common Error:

Poor TRANSLATE reasoning: Students make sign errors when substituting into the slope formula, especially with the denominator calculation. They might calculate \(\mathrm{4 - (-2)}\) as 2 instead of 6, giving them an incorrect slope.

This may lead them to select Choice D \(\mathrm{(\frac{-3}{2})}\) if they get \(\mathrm{m = \frac{-8}{2} = -4}\), then incorrectly simplify to \(\mathrm{\frac{-3}{2}}\).

The Bottom Line:

Success requires careful attention to negative signs and fraction reduction. The conceptual understanding is straightforward, but execution errors in basic algebra derail many students.