In the xy-plane, line l passes through the point \((0, 0)\) and is perpendicular to the line represented by the...
GMAT Algebra : (Alg) Questions
In the xy-plane, line l passes through the point \((0, 0)\) and is perpendicular to the line represented by the equation \(\mathrm{y = -2x + 7}\). If line l also passes through the point \((4, \mathrm{k})\), what is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem information
- Given information:
- Line l passes through origin \(\mathrm{(0, 0)}\)
- Line l is perpendicular to \(\mathrm{y = -2x + 7}\)
- Line l passes through point \(\mathrm{(4, k)}\)
- What we need: Find the value of k
2. INFER the approach
- To find k, we need line l's equation
- To get line l's equation, we need its slope
- Since l is perpendicular to \(\mathrm{y = -2x + 7}\), we can find l's slope using the perpendicular slope relationship
3. TRANSLATE the slope from the given line
- The line \(\mathrm{y = -2x + 7}\) has slope \(\mathrm{-2}\) (coefficient of x)
4. INFER the slope of line l
- Perpendicular lines have slopes that are negative reciprocals
- Slope of line l = negative reciprocal of \(\mathrm{-2}\) = \(\mathrm{-1/(-2) = 1/2}\)
5. INFER the equation of line l
- Line l passes through origin \(\mathrm{(0, 0)}\) with slope \(\mathrm{1/2}\)
- Equation: \(\mathrm{y = (1/2)x}\)
6. SIMPLIFY to find k
- Line l passes through \(\mathrm{(4, k)}\), so substitute \(\mathrm{x = 4}\):
- \(\mathrm{k = (1/2)(4) = 2}\)
Answer: 2
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students confuse the perpendicular slope relationship and think perpendicular lines have slopes that are just negatives of each other (instead of negative reciprocals).
They calculate: slope of line l = \(\mathrm{-(-2) = 2}\), leading to equation \(\mathrm{y = 2x}\), then \(\mathrm{k = 2(4) = 8}\).
This may lead them to select an incorrect answer or get confused when 8 isn't among typical answer choices.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that the slope should be \(\mathrm{1/2}\), but make arithmetic errors either in finding the negative reciprocal or in the final multiplication.
Common mistakes: calculating \(\mathrm{-1/(-2)}\) incorrectly, or errors in \(\mathrm{(1/2)(4)}\). This leads to various incorrect values for k and potential guessing.
The Bottom Line:
Success requires both understanding the geometric relationship between perpendicular lines AND executing the algebraic steps correctly. The perpendicular slope relationship is the key conceptual hurdle most students miss.