In the xy-plane, line k has a slope of 5 and a y-intercept of \((0, -35)\). What is the x-coordinate...

1. TRANSLATE the given information into equation form

• Given information:
  • Slope = 5
  • Y-intercept = (0, -35), so b = -35

• Using slope-intercept form \(\mathrm{y = mx + b}\):
  \(\mathrm{y = 5x + (-35) = 5x - 35}\)

2. INFER what x-intercept means

• The x-intercept is where the line crosses the x-axis
• At this point, the y-coordinate equals 0
• So we need to solve: \(\mathrm{y = 0}\)

3. SIMPLIFY by solving the equation

• Substitute \(\mathrm{y = 0}\) into our equation:
  \(\mathrm{0 = 5x - 35}\)

• Add 35 to both sides:
  \(\mathrm{35 = 5x}\)

• Divide both sides by 5:
  \(\mathrm{x = 7}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about intercepts: Students mix up x-intercept and y-intercept concepts.

They might think the x-intercept is somehow related to the given y-intercept point (0, -35), perhaps thinking the x-coordinate of the x-intercept should be 0 or -35. This fundamental misunderstanding of what intercepts represent causes them to avoid the correct algebraic approach entirely.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{0 = 5x - 35}\) but make arithmetic errors.

Common mistakes include: forgetting to add 35 to both sides, incorrectly getting \(\mathrm{-35 = 5x}\) instead of \(\mathrm{35 = 5x}\), or dividing incorrectly. For instance, they might calculate \(\mathrm{35 ÷ 5 = -7}\) instead of 7, or make sign errors throughout.

This may lead them to select incorrect numerical answers.

The Bottom Line:

This problem tests whether students truly understand that intercepts are found by setting the other variable to zero, not just by memorizing intercept formulas. The arithmetic is straightforward once the concept is clear.