y = 4x + 14y = 15x - 8The solution to the given system of equations is \(\mathrm{(x, y)}\). What...
GMAT Algebra : (Alg) Questions
\(\mathrm{y = 4x + 1}\)
\(\mathrm{4y = 15x - 8}\)
The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of \(\mathrm{x - y}\)?
1. TRANSLATE the problem information
- Given system:
- \(\mathrm{y = 4x + 1}\) (Equation 1)
- \(\mathrm{4y = 15x - 8}\) (Equation 2)
- Find: The value of \(\mathrm{x - y}\)
2. INFER the solution strategy
- Since Equation 1 already gives us y in terms of x, substitution is the most direct approach
- We'll substitute this expression for y into Equation 2 to find x
3. SIMPLIFY by substituting and solving
- Substitute \(\mathrm{y = 4x + 1}\) into \(\mathrm{4y = 15x - 8}\):
\(\mathrm{4(4x + 1) = 15x - 8}\) - Apply distributive property:
\(\mathrm{16x + 4 = 15x - 8}\) - Collect like terms:
\(\mathrm{16x - 15x = -8 - 4}\)
\(\mathrm{x = -12}\)
4. SIMPLIFY to find y
- Substitute \(\mathrm{x = -12}\) back into \(\mathrm{y = 4x + 1}\):
\(\mathrm{y = 4(-12) + 1 = -48 + 1 = -47}\)
5. SIMPLIFY to calculate the final answer
- \(\mathrm{x - y = -12 - (-47) = -12 + 47 = 35}\)
Answer: 35
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Sign error when calculating \(\mathrm{x - y = -12 - (-47)}\)
Students often struggle with subtracting negative numbers, writing \(\mathrm{x - y = -12 - 47 = -59}\) instead of correctly handling the double negative to get \(\mathrm{-12 + 47 = 35}\). This fundamental arithmetic error completely changes the final answer.
Second Most Common Error:
Poor INFER reasoning: Attempting elimination method instead of substitution
Some students don't recognize the advantage of having y already isolated in the first equation. They might try to eliminate variables by manipulating both equations unnecessarily, leading to more complex calculations and increased chance of arithmetic errors.
The Bottom Line:
This problem tests both strategic thinking (recognizing when substitution is optimal) and careful arithmetic execution, especially with negative numbers. The solution path is straightforward once the method is chosen, but precision in algebraic manipulation is crucial.