A line in the xy-plane has a slope of 1/3 and passes through the point \((3, 5)\). Which equation represents...

1. TRANSLATE the problem information

• Given information:
  • Slope: \(\mathrm{m = \frac{1}{3}}\)
  • Point on the line: \(\mathrm{(3, 5)}\)
  • Need to find: equation in slope-intercept form \(\mathrm{y = mx + b}\)
• What this tells us: We have a slope and one specific point, so point-slope form is the most direct approach.

2. INFER the approach

• Since we know the slope and one point, use point-slope form: \(\mathrm{y - y_1 = m(x - x_1)}\)
• Then convert to slope-intercept form to match the answer choices
• Substitute our known values: \(\mathrm{m = \frac{1}{3}, x_1 = 3, y_1 = 5}\)

3. SIMPLIFY using point-slope form

Set up the equation:
\(\mathrm{y - 5 = \frac{1}{3}(x - 3)}\)

Distribute the \(\mathrm{\frac{1}{3}}\):
\(\mathrm{y - 5 = \frac{1}{3}x - 1}\)

Add 5 to both sides:
\(\mathrm{y = \frac{1}{3}x - 1 + 5}\)
\(\mathrm{y = \frac{1}{3}x + 4}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{y - 5 = \frac{1}{3}(x - 3)}\) but make arithmetic errors when distributing.

Common mistake: \(\mathrm{\frac{1}{3}(x - 3) = \frac{1}{3}x - 3}\) instead of \(\mathrm{\frac{1}{3}x - 1}\)

This leads to: \(\mathrm{y = \frac{1}{3}x - 3 + 5 = \frac{1}{3}x + 2}\)

This may lead them to select Choice B (\(\mathrm{y = \frac{1}{3}x + 2}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse the slope sign or substitute incorrectly into point-slope form.

Some students might think the slope should be negative or mix up the coordinates, leading to incorrect setups that don't match any systematic approach.

This leads to confusion and guessing between the remaining choices.

The Bottom Line:

This problem tests whether students can systematically apply point-slope form and execute the algebraic steps carefully. The key is methodical substitution and precise arithmetic - small errors in distribution or combining terms lead directly to wrong answer choices that look very similar to the correct one.