In the xy-plane, points \(\mathrm{A(1, 5)}\) and \(\mathrm{B(4, 14)}\) determine line p. Line ell passes through the origin \(\mathrm{(0, 0)}\)...
GMAT Algebra : (Alg) Questions
In the xy-plane, points \(\mathrm{A(1, 5)}\) and \(\mathrm{B(4, 14)}\) determine line \(\mathrm{p}\). Line \(\mathrm{\ell}\) passes through the origin \(\mathrm{(0, 0)}\) and is parallel to line \(\mathrm{p}\). If point \(\mathrm{P}\) has \(\mathrm{x}\)-coordinate \(\mathrm{-2}\) and lies on line \(\mathrm{\ell}\), what is the \(\mathrm{y}\)-coordinate of \(\mathrm{P}\)?
Answer Format: Enter an integer.
Fill-in-the-blank (no choices)
1. TRANSLATE the problem information
- Given information:
- Points \(\mathrm{A(1, 5)}\) and \(\mathrm{B(4, 14)}\) are on line p
- Line ℓ passes through origin \(\mathrm{(0, 0)}\)
- Line ℓ is parallel to line p
- Point P is on line ℓ with x-coordinate = \(\mathrm{-2}\)
- What we need to find: y-coordinate of point P
2. INFER the solution strategy
- To find point P on line ℓ, we need the equation of line ℓ
- To get the equation of line ℓ, we need its slope
- Since line ℓ is parallel to line p, we can find the slope by first calculating the slope of line p
3. SIMPLIFY to find the slope of line p
Using the slope formula with \(\mathrm{A(1, 5)}\) and \(\mathrm{B(4, 14)}\):
- \(\mathrm{m = \frac{14 - 5}{4 - 1}}\)
- \(\mathrm{m = \frac{9}{3}}\)
- \(\mathrm{m = 3}\)
4. INFER the slope and equation of line ℓ
- Since parallel lines have the same slope: slope of line ℓ \(\mathrm{= 3}\)
- Since line ℓ passes through the origin \(\mathrm{(0, 0)}\): \(\mathrm{y = 3x}\)
5. SIMPLIFY to find the y-coordinate of point P
- Substitute \(\mathrm{x = -2}\) into \(\mathrm{y = 3x}\)
- \(\mathrm{y = 3(-2)}\)
- \(\mathrm{y = -6}\)
Answer: -6
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that parallel lines have the same slope, or they forget that a line through the origin has the simple form \(\mathrm{y = mx}\) rather than \(\mathrm{y = mx + b}\).
Some students might try to find the equation of line p first (like \(\mathrm{y - 5 = 3(x - 1)}\)), then struggle to connect this to line ℓ. Others might write \(\mathrm{y = 3x + b}\) and waste time trying to find b, not realizing that \(\mathrm{b = 0}\) since the line passes through the origin.
This leads to confusion and potentially incorrect setup of the final equation.
Second Most Common Error:
Poor SIMPLIFY execution: Students make arithmetic errors, either when calculating the slope (getting \(\mathrm{9/3}\) wrong) or when substituting \(\mathrm{x = -2}\) (getting the sign wrong and calculating \(\mathrm{3(-2) = 6}\) instead of \(\mathrm{-6}\)).
This type of error typically results from rushing through the computational steps after correctly setting up the problem.
The Bottom Line:
This problem tests whether students can connect the concept of parallel lines to coordinate geometry. The key insight is recognizing that finding the slope of one line gives you the slope of any parallel line, and that lines through the origin have particularly simple equations.