Line p is defined by 2y + 18x = 9. Line r is perpendicular to line p in the xy-plane....
GMAT Algebra : (Alg) Questions
Line p is defined by \(2\mathrm{y} + 18\mathrm{x} = 9\). Line r is perpendicular to line p in the \(\mathrm{xy}\)-plane. What is the slope of line r?
1. TRANSLATE the given information
- Given: Line p is defined by \(\mathrm{2y + 18x = 9}\)
- Given: Line r is perpendicular to line p
- Find: slope of line r
2. TRANSLATE to find the slope of line p
- To find slope, convert equation to slope-intercept form \(\mathrm{y = mx + b}\)
- Starting with: \(\mathrm{2y + 18x = 9}\)
- Subtract 18x from both sides: \(\mathrm{2y = -18x + 9}\)
- Divide everything by 2: \(\mathrm{y = -9x + \frac{9}{2}}\)
- The slope of line p is \(\mathrm{-9}\)
3. INFER the relationship for perpendicular lines
- Perpendicular lines have slopes that are negative reciprocals
- If one line has slope \(\mathrm{m}\), the perpendicular line has slope \(\mathrm{-\frac{1}{m}}\)
- Since line p has slope \(\mathrm{-9}\), line r has slope \(\mathrm{-\frac{1}{(-9)}}\)
4. SIMPLIFY the negative reciprocal calculation
- \(\mathrm{-\frac{1}{(-9)} = \frac{1}{9}}\)
- Therefore, slope of line r = \(\mathrm{\frac{1}{9}}\)
Answer: C. \(\mathrm{\frac{1}{9}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students incorrectly identify the slope from standard form equation
Many students see \(\mathrm{2y + 18x = 9}\) and think the slope is either 18 or 2, not realizing they need to convert to \(\mathrm{y = mx + b}\) form first. They might think "the coefficient of x is the slope" without doing the algebraic manipulation.
This may lead them to select Choice D (9) if they use 18 as slope and take its reciprocal, or get confused and guess.
Second Most Common Error:
Conceptual confusion about perpendicular slopes: Students forget about the "negative" part of negative reciprocal
Students correctly find that line p has slope \(\mathrm{-9}\), but then think perpendicular lines just have reciprocal slopes (not negative reciprocal). So they calculate \(\mathrm{\frac{1}{(-9)} = -\frac{1}{9}}\) instead of \(\mathrm{-\frac{1}{(-9)} = \frac{1}{9}}\).
This may lead them to select Choice B \(\mathrm{(-\frac{1}{9})}\).
The Bottom Line:
This problem tests whether students can systematically work with linear equations in different forms and apply the perpendicular line relationship correctly - both conceptual knowledge and careful algebraic execution are essential.