The table shows three values of x and their corresponding values of y, where n is a constant, for the...
GMAT Algebra : (Alg) Questions
The table shows three values of \(\mathrm{x}\) and their corresponding values of \(\mathrm{y}\), where \(\mathrm{n}\) is a constant, for the linear relationship between \(\mathrm{x}\) and \(\mathrm{y}\). What is the slope of the line that represents this relationship in the \(\mathrm{xy}\)-plane?
| \(\mathrm{x}\) | \(\mathrm{y}\) |
|---|---|
| \(\mathrm{-6}\) | \(\mathrm{n + 184}\) |
| \(\mathrm{-3}\) | \(\mathrm{n + 92}\) |
| \(\mathrm{0}\) | \(\mathrm{n}\) |
\(-\frac{92}{3}\)
\(-\frac{3}{92}\)
\(\frac{\mathrm{n}+92}{3}\)
\(\frac{2\mathrm{n}-92}{3}\)
1. TRANSLATE the table information
- Given information:
- Three points on a linear relationship: \((-6, n + 184), (-3, n + 92), (0, n)\)
- Need to find the slope of this line
- What this tells us: We have coordinate pairs where n is a constant that appears in each y-value
2. INFER the approach
- For linear relationships, slope is constant between any two points
- We can use the slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
- Strategic choice: Use points where the calculation will be cleanest
3. TRANSLATE chosen points into slope formula
- Using points \((-3, n + 92)\) and \((0, n)\):
- \(\mathrm{m = \frac{n - (n + 92)}{0 - (-3)}}\)
4. SIMPLIFY the algebraic expression
- Numerator: \(\mathrm{n - (n + 92) = n - n - 92 = -92}\)
- Denominator: \(\mathrm{0 - (-3) = 3}\)
- Therefore: \(\mathrm{m = -\frac{92}{3}}\)
Answer: A. \(\mathrm{-\frac{92}{3}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make algebraic errors when handling \(\mathrm{n - (n + 92)}\)
Many students incorrectly simplify this as:
- \(\mathrm{n - (n + 92) = -92 - n}\) (forgetting to distribute the negative)
- Or \(\mathrm{n - (n + 92) = 0 + 92 = 92}\) (thinking n - n = 0, ignoring the -92)
This leads them to calculate slopes like \(\mathrm{\frac{92}{3}}\) or \(\mathrm{\frac{92-n}{3}}\), causing confusion and potentially selecting Choice C (\(\mathrm{\frac{n+92}{3}}\)) or guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students confuse which values are x-coordinates versus y-coordinates or invert the slope formula
They might calculate \(\mathrm{\frac{x_2 - x_1}{y_2 - y_1}}\) instead, getting \(\mathrm{-\frac{3}{92}}\), leading them to select Choice B (\(\mathrm{-\frac{3}{92}}\)).
The Bottom Line:
This problem tests whether students can work systematically with variables in coordinate geometry. The key insight is recognizing that even though n is unknown, it cancels out perfectly in the slope calculation, yielding a clean numerical answer.
\(-\frac{92}{3}\)
\(-\frac{3}{92}\)
\(\frac{\mathrm{n}+92}{3}\)
\(\frac{2\mathrm{n}-92}{3}\)