A line in the xy-plane has a slope of 1/9 and passes through the point \((0, 14)\). Which equation represents...

1. TRANSLATE the problem information

• Given information:
  • Slope = \(\frac{1}{9}\)
  • Line passes through point \(\mathrm{(0, 14)}\)
• What this tells us: We have the slope directly, and the point \(\mathrm{(0, 14)}\) is special because it's on the y-axis

2. INFER the approach

• Since we have slope and a point, we can use slope-intercept form: \(\mathrm{y = mx + b}\)
• The point \(\mathrm{(0, 14)}\) is particularly useful because it's already in the form \(\mathrm{(0, b)}\), which means \(\mathrm{b = 14}\)
• This makes our job easier - we don't need to do any additional calculations

3. TRANSLATE the components into the formula

• In \(\mathrm{y = mx + b}\):
  • \(\mathrm{m = \frac{1}{9}}\) (the given slope)
  • \(\mathrm{b = 14}\) (from the point \(\mathrm{(0, 14)}\))

4. Write the final equation

• Substituting: \(\mathrm{y = \frac{1}{9}x + 14}\)

Answer: D. \(\mathrm{y = \frac{1}{9}x + 14}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the signs in the slope or mix up which number is the slope versus the y-intercept.

They might think the slope should be negative or accidentally use -14 for the y-intercept. This confusion with signs leads them to select Choice A (\(\mathrm{y = -\frac{1}{9}x - 14}\)) or Choice B (\(\mathrm{y = -\frac{1}{9}x + 14}\)).

Second Most Common Error:

Conceptual confusion about y-intercept: Students don't recognize that when a line "passes through \(\mathrm{(0, 14)}\)," this directly gives them the y-intercept.

Instead, they try to use point-slope form or substitute \(\mathrm{(0, 14)}\) into their equation to "solve for b," creating unnecessary work and potential calculation errors. This may lead them to select Choice C (\(\mathrm{y = \frac{1}{9}x - 14}\)) if they make a sign error during their extra calculations.

The Bottom Line:

This problem rewards students who recognize the shortcut: when given a point of the form \(\mathrm{(0, b)}\), that's your y-intercept handed to you on a silver platter. The key insight is recognizing what information you already have versus what you need to calculate.