The table above shows the coordinates of three points on a line in the xy-plane, where k and n are...
GMAT Algebra : (Alg) Questions
The table above shows the coordinates of three points on a line in the \(\mathrm{xy}\)-plane, where \(\mathrm{k}\) and \(\mathrm{n}\) are constants. If the slope of the line is \(2\), what is the value of \(\mathrm{k + n}\)?
| \(\mathrm{x}\) | \(\mathrm{y}\) |
|---|---|
| \(3\) | \(7\) |
| \(\mathrm{k}\) | \(11\) |
| \(12\) | \(\mathrm{n}\) |
1. TRANSLATE the problem information
- Given information:
- Three points on a line: \((3, 7), (k, 11), (12, n)\)
- The slope of the line is 2
- Need to find \(k + n\)
2. INFER the solution approach
- Since all three points lie on the same line with slope 2, we can use the slope formula between any two points
- Strategy: Use the slope formula twice - once to find k, once to find n
- Key insight: \(\mathrm{slope} = \frac{y_2 - y_1}{x_2 - x_1} = 2\) for any pair of points on this line
3. SIMPLIFY to find k using points (3, 7) and (k, 11)
- Set up the slope equation: \(\frac{11 - 7}{k - 3} = 2\)
- Simplify the numerator: \(\frac{4}{k - 3} = 2\)
- Multiply both sides by \((k - 3)\): \(4 = 2(k - 3)\)
- Distribute: \(4 = 2k - 6\)
- Add 6 to both sides: \(10 = 2k\)
- Divide by 2: \(k = 5\)
4. SIMPLIFY to find n using points (3, 7) and (12, n)
- Set up the slope equation: \(\frac{n - 7}{12 - 3} = 2\)
- Simplify the denominator: \(\frac{n - 7}{9} = 2\)
- Multiply both sides by 9: \(n - 7 = 18\)
- Add 7 to both sides: \(n = 25\)
5. Calculate the final answer
- \(k + n = 5 + 25 = 30\)
Answer: 30
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students may not recognize that they can use any two points on the line to apply the slope formula. Instead, they might try to use all three points simultaneously in a single equation, leading to confusion about how to set up the problem systematically.
This leads to abandoning systematic solution and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up the slope equations but make algebraic errors, such as:
- Forgetting to distribute the 2 in \(2(k - 3)\), getting \(4 = 2k - 3\) instead of \(4 = 2k - 6\)
- Sign errors when rearranging equations
- Calculation mistakes in the final arithmetic
These errors could lead to incorrect values for k or n, resulting in wrong answers like 20, 25, or other values depending on the specific mistake.
The Bottom Line:
This problem tests whether students can systematically apply the slope formula multiple times and maintain accuracy through multi-step algebra. The key insight is recognizing that the same slope applies to any pair of points on the line.