In the xy-plane, line m passes through the points \((0, 5)\) and \((2, 11)\). Which equation defines line m?y =...

1. TRANSLATE the problem information

• Given information:
  • Line m passes through points \(\mathrm{(0, 5)}\) and \(\mathrm{(2, 11)}\)
  • Need to find the equation that defines line m
• We need to find an equation in the form \(\mathrm{y = mx + b}\)

2. INFER the approach

• To write a linear equation, we need two key components:
  • The slope (m)
  • The y-intercept (b)
• Strategy: Use the slope formula with our two points, then identify the y-intercept

3. SIMPLIFY to find the slope

• Using the slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substitute our points \(\mathrm{(0, 5)}\) and \(\mathrm{(2, 11)}\):

\(\mathrm{m = \frac{11 - 5}{2 - 0}}\)
\(\mathrm{m = \frac{6}{2}}\)
\(\mathrm{m = 3}\)

4. INFER the y-intercept

• The y-intercept occurs when \(\mathrm{x = 0}\)
• Since we have the point \(\mathrm{(0, 5)}\), the y-intercept is 5

5. SIMPLIFY to write the final equation

• Using slope-intercept form: \(\mathrm{y = mx + b}\)
• Substitute \(\mathrm{m = 3}\) and \(\mathrm{b = 5}\): \(\mathrm{y = 3x + 5}\)

6. INFER verification strategy and check

• Test our equation with the second point \(\mathrm{(2, 11)}\):

\(\mathrm{y = 3(2) + 5}\)
\(\mathrm{y = 6 + 5}\)
\(\mathrm{y = 11}\) ✓

Answer: A (\(\mathrm{y = 3x + 5}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often confuse which point gives which information, particularly mixing up how to use the point \(\mathrm{(0, 5)}\). They might incorrectly think the y-intercept is from the point \(\mathrm{(2, 11)}\), leading to wrong equation setup.

This conceptual confusion about extracting the y-intercept may lead them to select Choice D (\(\mathrm{y = 3x + 11}\)) - they get the slope right but use the wrong y-intercept.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating the slope, often getting \(\mathrm{\frac{6}{2} = 2}\) instead of 3, or setting up the slope formula incorrectly with coordinates in wrong positions.

This calculation error may lead them to select Choice C (\(\mathrm{y = x + 5}\)) where the slope is wrong but the y-intercept is correct.

The Bottom Line:

This problem tests whether students can systematically extract slope and y-intercept information from coordinate pairs. The key insight is recognizing that when one point has \(\mathrm{x = 0}\), it directly gives you the y-intercept, making the problem more straightforward than it initially appears.