In the xy-plane, a system of two linear equations has no solution. If the first equation is 3x - y...
GMAT Algebra : (Alg) Questions
In the xy-plane, a system of two linear equations has no solution. If the first equation is \(3\mathrm{x} - \mathrm{y} = -5\) and the second equation is \(\mathrm{y} = \mathrm{kx} + 2\), where \(\mathrm{k}\) is a constant, what is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem requirements
- Given information:
- First equation: \(\mathrm{3x - y = -5}\)
- Second equation: \(\mathrm{y = kx + 2}\)
- System has no solution
- What this tells us: We need to find the value of k that makes these lines parallel
2. INFER the mathematical condition
- "No solution" means the lines are parallel but not identical
- Parallel lines have the same slope but different y-intercepts
- Strategy: Find the slope of the first equation and set k equal to it
3. SIMPLIFY the first equation to slope-intercept form
Starting with \(\mathrm{3x - y = -5}\):
- Move the x term: \(\mathrm{-y = -3x - 5}\)
- Multiply by -1: \(\mathrm{y = 3x + 5}\)
- The slope is 3, y-intercept is 5
4. INFER the slope relationship
- Second equation \(\mathrm{y = kx + 2}\) has slope k and y-intercept 2
- For parallel lines: \(\mathrm{k = 3}\)
5. Verify the solution
- First line: \(\mathrm{y = 3x + 5}\) (slope = 3, y-intercept = 5)
- Second line: \(\mathrm{y = 3x + 2}\) (slope = 3, y-intercept = 2)
- Same slopes but different y-intercepts ✓ No solution confirmed
Answer: C) 3
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when converting \(\mathrm{3x - y = -5}\) to slope-intercept form
They might get:
- \(\mathrm{y = -3x + 5}\) (wrong slope of -3)
- \(\mathrm{y = 3x - 5}\) (wrong y-intercept)
If they get slope = -3, they conclude \(\mathrm{k = -3}\), leading them to select Choice A (-3).
Second Most Common Error:
Poor INFER reasoning: Students don't properly understand what "no solution" means for a system
They might think "no solution" means the coefficients should be different or try to solve the system algebraically and get confused when they reach a contradiction like \(\mathrm{5 = 2}\).
This leads to confusion and guessing among the answer choices.
The Bottom Line:
Success requires both careful algebra (converting to slope-intercept form without sign errors) and conceptual understanding that parallel lines create systems with no solution.