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

1. TRANSLATE the problem information

• Given information: A table with x-values (1, 2, 3) and corresponding y-values (11, 16, 21)
• This gives us coordinate points: \(\mathrm{(1, 11), (2, 16), (3, 21)}\)
• We need to find the linear equation in the form \(\mathrm{y = mx + b}\)

2. INFER the solution strategy

• Since we have a linear relationship, we need to find two things:
  • The slope (m)
  • The y-intercept (b)
• Strategy: Use any two points to calculate slope, then use one point to find y-intercept

3. SIMPLIFY to find the slope

• Using points \(\mathrm{(1, 11)}\) and \(\mathrm{(2, 16)}\) with the slope formula:
  \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
  \(\mathrm{m = \frac{16 - 11}{2 - 1}}\)
  \(\mathrm{m = \frac{5}{1}}\)
  \(\mathrm{m = 5}\)

4. SIMPLIFY to find the y-intercept

• Substitute slope \(\mathrm{m = 5}\) and point \(\mathrm{(1, 11)}\) into \(\mathrm{y = mx + b}\):
  \(\mathrm{11 = 5(1) + b}\)
  \(\mathrm{11 = 5 + b}\)
  \(\mathrm{b = 6}\)

5. Write and verify the equation

• The equation is \(\mathrm{y = 5x + 6}\)
• Quick check with point \(\mathrm{(3, 21)}\):
  \(\mathrm{y = 5(3) + 6}\)
  \(\mathrm{y = 15 + 6}\)
  \(\mathrm{y = 21}\) ✓

Answer: A. \(\mathrm{y = 5x + 6}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the slope or solving for the y-intercept. For example, they might incorrectly calculate \(\mathrm{\frac{16-11}{2-1}}\) as 6 instead of 5, or make an error when solving \(\mathrm{11 = 5 + b}\).

This may lead them to select Choice C (\(\mathrm{y = 6x + 5}\)) or Choice D (\(\mathrm{y = 6x + 11}\)).

Second Most Common Error:

Poor INFER reasoning: Students understand they need \(\mathrm{y = mx + b}\) but don't realize they need to find the y-intercept after finding the slope. They might assume the y-intercept is simply the first y-value (11) without doing the algebra.

This may lead them to select Choice B (\(\mathrm{y = 5x + 11}\)).

The Bottom Line:

This problem tests whether students can systematically work through the two-step process of finding a linear equation: calculate slope first, then use that slope with a point to find the y-intercept. Success requires careful arithmetic and understanding the logical sequence of steps.