3x + 6 = 4y 3x + 4 = 2y The solution to the given system of equations is \(\mathrm{(x,...
GMAT Algebra : (Alg) Questions
\(3\mathrm{x} + 6 = 4\mathrm{y}\)
\(3\mathrm{x} + 4 = 2\mathrm{y}\)
The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of y?
1. TRANSLATE the problem information
- Given system:
- Equation 1: \(\mathrm{3x + 6 = 4y}\)
- Equation 2: \(\mathrm{3x + 4 = 2y}\)
- Find: The value of y
2. INFER the best solution approach
- Notice both equations have the same \(\mathrm{3x}\) term
- This makes elimination method ideal - we can subtract one equation from the other to eliminate x immediately
- Choose to subtract equation 2 from equation 1
3. SIMPLIFY through equation subtraction
- Set up the subtraction:
\(\mathrm{(3x + 6) - (3x + 4) = 4y - 2y}\) - Left side: \(\mathrm{3x + 6 - 3x - 4 = 0 + 2 = 2}\)
- Right side: \(\mathrm{4y - 2y = 2y}\)
- Result: \(\mathrm{2 = 2y}\)
4. SIMPLIFY to find the final answer
- Divide both sides by 2: \(\mathrm{2 ÷ 2 = 2y ÷ 2}\)
- Therefore: \(\mathrm{y = 1}\)
Answer: 1
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when subtracting the second equation from the first.
They might incorrectly calculate \(\mathrm{(3x + 6) - (3x + 4)}\) as \(\mathrm{3x + 6 - 3x + 4 = 10}\), missing the negative sign in front of the 4. This leads to \(\mathrm{10 = 2y}\), giving \(\mathrm{y = 5}\). Since this problem is student-response format, this leads to an incorrect numerical answer.
Second Most Common Error:
Poor INFER reasoning about method selection: Students attempt substitution instead of recognizing the elimination opportunity.
They might solve the first equation for x: \(\mathrm{x = \frac{4y - 6}{3}}\), then substitute into the second equation, creating a more complex fractional equation. While this approach can work, it's more error-prone and time-consuming, leading to arithmetic mistakes or abandoning the systematic solution.
The Bottom Line:
This problem rewards students who can quickly spot structural similarities in equations and choose the most efficient solution path. The key insight is recognizing that identical terms (\(\mathrm{3x}\) in both equations) signal an immediate elimination opportunity.