In the xy-plane, a line has a slope of 6 and passes through the point \((0,8)\). Which of the following...

1. TRANSLATE the problem information

• Given information:
  • Slope = 6
  • Line passes through point \(\mathrm{(0, 8)}\)

• We need to find the equation of this line

2. INFER the approach

• Since we have slope and a point, we can use slope-intercept form: \(\mathrm{y = mx + b}\)
• The key insight: the point \(\mathrm{(0, 8)}\) is special because when \(\mathrm{x = 0}\), we're at the y-intercept
• This means \(\mathrm{b = 8}\) directly!

3. INFER the values for the slope-intercept form

• In \(\mathrm{y = mx + b}\):
  • \(\mathrm{m = 6}\) (the given slope)
  • \(\mathrm{b = 8}\) (since the line passes through \(\mathrm{(0, 8)}\), this is the y-intercept)

4. Write the final equation

• Substituting into \(\mathrm{y = mx + b}\):
  \(\mathrm{y = 6x + 8}\)

Answer: A. \(\mathrm{y = 6x + 8}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about slope vs y-intercept: Students know both values are 6 and 8, but mix up which is which in the slope-intercept form.

They might think: "The slope is 6 and it passes through \(\mathrm{(0, 8)}\), so maybe the slope goes with x and the 8 goes with... wait, which is which?" This confusion leads them to swap the values.

This may lead them to select Choice C (\(\mathrm{y = 8x + 6}\))

Second Most Common Error:

Weak INFER skill: Students don't recognize that \(\mathrm{(0, 8)}\) directly gives the y-intercept and instead try to use more complicated approaches like point-slope form, potentially making calculation errors along the way.

This might cause them to get confused during calculations and guess, or make arithmetic mistakes that don't match any of the clean answer choices.

The Bottom Line:

The beauty of this problem is recognizing that when you're given a point where \(\mathrm{x = 0}\), you immediately know the y-intercept. Students who miss this insight make the problem much harder than it needs to be.