The table shows three values of x and their corresponding values of y. Which equation represents the linear relationship between...

1. TRANSLATE the table information into coordinate points

• Given information:
  • Point 1: \((-1, 8)\)
  • Point 2: \((0, 5)\)
  • Point 3: \((1, 2)\)

2. INFER what components you need for y = mx + b

• You need to find slope (m) and y-intercept (b)
• The y-intercept is easy to spot: when \(\mathrm{x = 0}\), \(\mathrm{y = 5}\), so \(\mathrm{b = 5}\)

3. SIMPLIFY the slope calculation using any two points

• Using points \((0, 5)\) and \((1, 2)\):
• \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{2 - 5}{1 - 0}}\)
• \(\mathrm{m = \frac{-3}{1}}\)
• \(\mathrm{m = -3}\)

4. Form the equation and verify

• Equation: \(\mathrm{y = -3x + 5}\)
• SIMPLIFY verification with third point \((-1, 8)\):

\(\mathrm{y = -3(-1) + 5}\)

\(\mathrm{y = 3 + 5}\)

\(\mathrm{y = 8}\) ✓

Answer: A (y = -3x + 5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when calculating slope, especially with negative differences

Students often calculate \(\mathrm{m = \frac{2 - 5}{1 - 0} = \frac{-3}{1} = +3}\) (dropping the negative sign)

This leads them to think the equation is \(\mathrm{y = 3x + 5}\) and select Choice B (y = 3x + 5)

Second Most Common Error:

Poor INFER reasoning: Not recognizing which table value represents the y-intercept

Students might get confused about finding the y-intercept and use the wrong constant term

This could lead them to select Choice D (y = -3x - 5) if they get the slope right but use the wrong y-intercept

The Bottom Line:

This problem tests whether students can systematically extract slope and y-intercept from tabular data. The key insight is recognizing that the y-intercept jumps out immediately when \(\mathrm{x = 0}\), and then carefully handling the arithmetic signs during slope calculation.