For line h, the table shows three values of x and their corresponding values of y. Line k is the...

1. TRANSLATE the problem information

• Given information:
  • Three points on line h: \((18, 130)\), \((23, 160)\), \((26, 178)\)
  • Line k is line h translated down 5 units
  • Need to find: x-intercept of line k

2. INFER the solution strategy

• To find line k's x-intercept, I need line k's equation first
• To get line k's equation, I need line h's equation first
• Strategy: Find line h → Apply translation → Find x-intercept

3. SIMPLIFY to find line h's equation

• Calculate slope using any two points:
  \(\mathrm{slope = \frac{160 - 130}{23 - 18} = \frac{30}{5} = 6}\)

• Use \(\mathrm{y = mx + b}\) with \(\mathrm{slope = 6}\):
  \(\mathrm{130 = 6(18) + b}\)
  \(\mathrm{130 = 108 + b}\)
  \(\mathrm{b = 22}\)

• Line h equation: \(\mathrm{y = 6x + 22}\)

4. TRANSLATE the translation into mathematics

• "Down 5 units" means subtract 5 from y-values
• Line k equation: \(\mathrm{y = 6x + 22 - 5 = 6x + 17}\)

5. SIMPLIFY to find the x-intercept

• At x-intercept, \(\mathrm{y = 0}\):
  \(\mathrm{0 = 6x + 17}\)
  \(\mathrm{-17 = 6x}\)
  \(\mathrm{x = \frac{-17}{6}}\)

Answer: D. \(\mathrm{(\frac{-17}{6}, 0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse "down 5 units" with other operations, thinking it means subtracting 5 from x-values instead of y-values, or adding instead of subtracting.

For example, they might create line k as \(\mathrm{y = 6x + 27}\) (adding 5 instead of subtracting), leading to x-intercept of \(\mathrm{\frac{-27}{6} = \frac{-9}{2}}\). This may lead them to select Choice B \(\mathrm{(\frac{-9}{2}, 0)}\).

Second Most Common Error:

Poor INFER reasoning: Students jump directly to finding an x-intercept without first establishing line h's complete equation. They might try shortcuts using just the slope and one point, leading to calculation errors and confusion about what equation they're actually working with.

The Bottom Line:

This problem tests whether students can systematically work through multi-step function transformations. The key insight is recognizing that you need the complete original equation before you can apply any transformation.