8x + 7y = 9 24x + 21y = 27 For each real number r, which of the following points...
GMAT Algebra : (Alg) Questions
\(8\mathrm{x} + 7\mathrm{y} = 9\)
\(24\mathrm{x} + 21\mathrm{y} = 27\)
For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?
\(\left(\mathrm{r}, -\frac{8\mathrm{r}}{7} + \frac{9}{7}\right)\)
\(\left(-\frac{8\mathrm{r}}{7} + \frac{9}{7}, \mathrm{r}\right)\)
\(\left(-\frac{8\mathrm{r}}{7} + 9, \frac{8\mathrm{r}}{7} + 27\right)\)
\(\left(\frac{\mathrm{r}}{8} + 9, -\frac{\mathrm{r}}{8} + 27\right)\)
1. INFER the relationship between the equations
- Given system:
- \(8x + 7y = 9\)
- \(24x + 21y = 27\)
- Key insight: Before solving, examine if these equations are related
- Look at the second equation: \(24x + 21y = 27\)
- Divide everything by 3: \(8x + 7y = 9\)
- This is identical to the first equation!
2. INFER what this means geometrically
- Since both equations represent the same line, they have infinitely many solutions
- Every point on this line satisfies both equations
- We need to find which answer choice represents points on this line
3. SIMPLIFY by testing the answer choices
- We only need to check if the points satisfy the equation \(8x + 7y = 9\)
- Testing choice A: \((r, -\frac{8r}{7} + \frac{9}{7})\)
- Substitute \(x = r\) and \(y = -\frac{8r}{7} + \frac{9}{7}\):
\(8r + 7(-\frac{8r}{7} + \frac{9}{7})\)
\(= 8r + 7(-\frac{8r}{7}) + 7(\frac{9}{7})\)
\(= 8r - 8r + 9\)
\(= 9\) ✓
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that the second equation is just 3 times the first equation, so they try to solve the system using elimination or substitution as if it has a unique solution.
When they can't find a unique (x,y) pair, they get confused about how to approach the answer choices. This leads to confusion and guessing among the parametric expressions.
Second Most Common Error:
Poor SIMPLIFY execution: Students recognize the equations are equivalent but make algebraic errors when substituting the parametric expressions from the answer choices into the original equation.
For example, they might incorrectly distribute: \(7(-\frac{8r}{7} + \frac{9}{7})\) and get \(7(-\frac{8r}{7}) + \frac{9}{7}\) instead of \(7(-\frac{8r}{7}) + 7(\frac{9}{7})\), leading to \(-8r + \frac{9}{7}\) instead of \(-8r + 9\). This may lead them to select Choice B or abandon systematic checking.
The Bottom Line:
This problem tests whether students can recognize when a system represents the same line (infinite solutions) rather than trying to force a unique solution approach. The key insight is seeing the equivalence relationship between the equations.
\(\left(\mathrm{r}, -\frac{8\mathrm{r}}{7} + \frac{9}{7}\right)\)
\(\left(-\frac{8\mathrm{r}}{7} + \frac{9}{7}, \mathrm{r}\right)\)
\(\left(-\frac{8\mathrm{r}}{7} + 9, \frac{8\mathrm{r}}{7} + 27\right)\)
\(\left(\frac{\mathrm{r}}{8} + 9, -\frac{\mathrm{r}}{8} + 27\right)\)