The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is...
GMAT Algebra : (Alg) Questions
The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is \(\mathrm{(k - 5, b)}\), where \(\mathrm{k}\) and \(\mathrm{b}\) are constants. What is the value of \(\mathrm{b}\)?
| x | y |
|---|---|
| \(\mathrm{k}\) | \(13\) |
| \(\mathrm{k} + 7\) | \(-15\) |
1. TRANSLATE the problem information
- Given information:
- Two points on the line: \(\mathrm{(k, 13)}\) and \(\mathrm{(k + 7, -15)}\)
- Y-intercept: \(\mathrm{(k - 5, b)}\) where b is unknown
- Need to find the value of b
2. INFER the approach
- Since all three points lie on the same line, the slope between any two points must be the same
- Strategy: Calculate slope using the two given points, then use that slope with the y-intercept point to find b
3. SIMPLIFY to find the slope
- Using points \(\mathrm{(k, 13)}\) and \(\mathrm{(k + 7, -15)}\):
- \(\mathrm{m} = \frac{-15 - 13}{\mathrm{k} + 7 - \mathrm{k}}\)
\(\mathrm{m} = \frac{-28}{7}\)
\(\mathrm{m} = -4\)
4. INFER the relationship for the y-intercept
- The slope between y-intercept \(\mathrm{(k - 5, b)}\) and point \(\mathrm{(k, 13)}\) must also equal -4
- \(\mathrm{m} = \frac{13 - \mathrm{b}}{\mathrm{k} - (\mathrm{k} - 5)}\)
\(\mathrm{m} = \frac{13 - \mathrm{b}}{5}\)
5. SIMPLIFY to solve for b
- Set the slopes equal: \(-4 = \frac{13 - \mathrm{b}}{5}\)
- Multiply both sides by 5: \(-20 = 13 - \mathrm{b}\)
- Subtract 13: \(-33 = -\mathrm{b}\)
- Therefore: \(\mathrm{b} = 33\)
Answer: 33
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students struggle to correctly identify and organize the coordinate points from the given information, especially interpreting what \(\mathrm{(k - 5, b)}\) means as the y-intercept.
They might confuse which coordinates represent which points or misunderstand that the y-intercept is also a point on the line. This leads to setting up incorrect slope equations or not knowing how to proceed systematically.
This leads to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up the equation \(-4 = \frac{13 - \mathrm{b}}{5}\) but make sign errors during algebraic manipulation.
Common mistakes include getting \(-20 = 13 - \mathrm{b}\) but then incorrectly solving to get \(\mathrm{b} = -33 - 13 = -46\), or other sign management errors when isolating b.
This may lead them to select incorrect numerical answers due to calculation mistakes.
The Bottom Line:
This problem requires students to recognize that the y-intercept, despite being described with variables, is still just another point on the line that must satisfy the same slope relationship as any other pair of points.