Line ell passes through the origin and has slope -{4/3}. Line m is perpendicular to line ell and passes through...
GMAT Algebra : (Alg) Questions
Line \(\ell\) passes through the origin and has slope \(-\frac{4}{3}\). Line m is perpendicular to line \(\ell\) and passes through the point \((0, 2)\). Which of the following defines the function g whose graph is line m?
- \(\mathrm{g(x)} = -\frac{4}{3}\mathrm{x} + 2\)
- \(\mathrm{g(x)} = \frac{3}{4}\mathrm{x} + 2\)
- \(\mathrm{g(x)} = \frac{4}{3}\mathrm{x} + 2\)
- \(\mathrm{g(x)} = \frac{3}{4}\mathrm{x} - 2\)
1. TRANSLATE the problem information
- Given information:
- Line ℓ: passes through origin, slope = \(-\frac{4}{3}\)
- Line m: perpendicular to line ℓ, passes through (0, 2)
- Find function g that represents line m
2. INFER the relationship between perpendicular lines
- Key insight: Perpendicular lines have slopes that are negative reciprocals
- Since line ℓ has slope \(-\frac{4}{3}\), line m has slope = \(-\frac{1}{-\frac{4}{3}}\)
- Strategy: Find slope of line m, then use point (0, 2) with slope-intercept form
3. SIMPLIFY to find the slope of line m
- Negative reciprocal of \(-\frac{4}{3}\):
- Take reciprocal: \(-\frac{4}{3}\) becomes \(-\frac{3}{4}\)
- Take negative: \(-(-\frac{3}{4}) = \frac{3}{4}\)
- Slope of line m = \(\frac{3}{4}\)
4. INFER the y-intercept from the given point
- Line m passes through (0, 2)
- When \(x = 0\), \(y = 2\), so y-intercept = 2
5. TRANSLATE into slope-intercept form
- Using \(y = mx + b\) where \(m = \frac{3}{4}\) and \(b = 2\)
- \(g(x) = \frac{3}{4}x + 2\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students confuse perpendicular with parallel lines and use the same slope (\(-\frac{4}{3}\)) instead of the negative reciprocal.
They think perpendicular means "similar" and incorrectly reason that both lines should have the same slope. This leads them to look for an answer with slope \(-\frac{4}{3}\).
This may lead them to select Choice A (\(g(x) = -\frac{4}{3}x + 2\)).
Second Most Common Error:
Poor SIMPLIFY execution: Students attempt to find the negative reciprocal but make calculation errors, getting \(\frac{4}{3}\) instead of \(\frac{3}{4}\).
Common mistakes include forgetting the negative sign or incorrectly flipping the fraction. They might calculate: negative reciprocal of \(-\frac{4}{3}\) = \(\frac{4}{3}\).
This may lead them to select Choice C (\(g(x) = \frac{4}{3}x + 2\)).
The Bottom Line:
This problem tests whether students truly understand the perpendicular slope relationship, not just memorization. The key insight is recognizing that "perpendicular" means negative reciprocal slopes, then executing the reciprocal calculation correctly.