Question:v/x = y - 9zThe given equation relates the numbers v, x, y, and z, where x is not equal...
GMAT Advanced Math : (Adv_Math) Questions
\(\frac{\mathrm{v}}{\mathrm{x}} = \mathrm{y} - 9\mathrm{z}\)
The given equation relates the numbers v, x, y, and z, where x is not equal to zero. Which equation correctly expresses v in terms of x, y, and z?
- \(\mathrm{v} = \frac{\mathrm{y} - 9\mathrm{z}}{\mathrm{x}}\)
- \(\mathrm{v} = \mathrm{x} + (\mathrm{y} - 9\mathrm{z})\)
- \(\mathrm{v} = \mathrm{y} - 9\mathrm{z} - \mathrm{x}\)
- \(\mathrm{v} = \mathrm{x}(\mathrm{y} - 9\mathrm{z})\)
1. TRANSLATE the problem information
- Given equation: \(\mathrm{v/x = y - 9z}\)
- Goal: Express v in terms of x, y, and z
2. INFER the solution strategy
- Currently v is divided by x
- To isolate v, I need to "undo" the division
- Since division and multiplication are inverse operations, I multiply both sides by x
3. SIMPLIFY by multiplying both sides by x
- Start with: \(\mathrm{v/x = y - 9z}\)
- Multiply both sides by x: \(\mathrm{(v/x) × x = (y - 9z) × x}\)
- On the left side, x cancels: \(\mathrm{v = x(y - 9z)}\)
Answer: D. \(\mathrm{v = x(y - 9z)}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that multiplication is the inverse of division, so they try incorrect operations like adding x to both sides.
Instead of multiplying by x, they might write: \(\mathrm{v/x + x = y - 9z + x}\), leading to confusion about how to isolate v. This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly decide to multiply by x but make algebraic errors in the process.
They might forget to distribute x to the entire right side, writing \(\mathrm{v = xy - 9z}\) instead of \(\mathrm{v = x(y - 9z)}\). This doesn't match any of the given choices, causing them to second-guess their approach and potentially select Choice A (\(\mathrm{v = (y - 9z)/x}\)), thinking they made an error in direction.
The Bottom Line:
This problem tests whether students understand inverse operations and can execute basic algebraic manipulation accurately. The key insight is recognizing that "undoing" division requires multiplication, not addition or subtraction.