Question:Line L1 passes through the points \(\left(\frac{1}{2}, 2\right)\) and \(\left(2, -\frac{1}{2}\right)\).Line L2 has equation y = tx + 3/2, whe...
GMAT Algebra : (Alg) Questions
Question:
- Line L1 passes through the points \(\left(\frac{1}{2}, 2\right)\) and \(\left(2, -\frac{1}{2}\right)\).
- Line L2 has equation \(y = \mathrm{t}x + \frac{3}{2}\), where \(\mathrm{t}\) is a constant.
- If these two distinct lines form a system with no solution, what is the value of \(\mathrm{t}\)?
Answer Format:
Express your answer as a fraction in lowest terms.
1. TRANSLATE the problem information
- Given information:
- L1 passes through points \(\left(\frac{1}{2}, 2\right)\) and \(\left(2, -\frac{1}{2}\right)\)
- L2 has equation \(\mathrm{y = tx + \frac{3}{2}}\)
- The system has no solution
- What this tells us: We need to find the value of t that makes this system impossible to solve
2. INFER the mathematical condition
- For a system of two linear equations to have no solution, the lines must be parallel but not identical
- This means: same slope, different y-intercepts
- Strategy: Find L1's slope, then set t equal to that slope
3. SIMPLIFY to find L1's slope
- Using the slope formula with points \(\left(\frac{1}{2}, 2\right)\) and \(\left(2, -\frac{1}{2}\right)\):
\(\mathrm{m = \frac{-\frac{1}{2} - 2}{2 - \frac{1}{2}} = \frac{-\frac{5}{2}}{\frac{3}{2}}}\)
- SIMPLIFY the complex fraction:
\(\left(-\frac{5}{2}\right) \div \left(\frac{3}{2}\right) = \left(-\frac{5}{2}\right) \times \left(\frac{2}{3}\right) = -\frac{5}{3}\)
4. APPLY the parallel condition
- L2's slope is t (from \(\mathrm{y = tx + \frac{3}{2}}\))
- For parallel lines: \(\mathrm{t = -\frac{5}{3}}\)
5. INFER verification needed
- We should verify the lines are truly distinct (different y-intercepts)
- Find L1's full equation using point-slope form with \(\left(\frac{1}{2}, 2\right)\):
\(\mathrm{y - 2 = \left(-\frac{5}{3}\right)\left(x - \frac{1}{2}\right)}\)
\(\mathrm{y = \left(-\frac{5}{3}\right)x + \frac{5}{6} + 2 = \left(-\frac{5}{3}\right)x + \frac{17}{6}}\)
6. APPLY CONSTRAINTS to confirm answer
- L1 has y-intercept: \(\frac{17}{6}\)
- L2 has y-intercept: \(\frac{3}{2} = \frac{9}{6}\)
- Since \(\frac{17}{6} \neq \frac{9}{6}\), the lines are parallel but distinct ✓
Answer: \(-\frac{5}{3}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't connect 'no solution' to the geometric relationship between lines. They might think 'no solution' means the equations can't be solved algebraically, leading them to set up incorrect equations or try to solve the system directly rather than analyzing line relationships.
This leads to confusion and guessing rather than systematic solution.
Second Most Common Error:
Poor SIMPLIFY execution: Students make arithmetic errors when computing the slope, especially with the complex fraction \(\frac{-\frac{5}{2}}{\frac{3}{2}}\). They might incorrectly get \(-\frac{5}{6}\) or \(\frac{5}{3}\) instead of \(-\frac{5}{3}\).
This may lead them to select an incorrect value of t.
The Bottom Line:
This problem requires students to think geometrically about linear systems rather than just algebraically. The key insight is translating 'no solution' into the precise condition of parallel but distinct lines.