Question: The ordered pair \(\mathrm{(x, y)}\) satisfies the system of equations y = x^2 + 4x + 12 and 2x...
GMAT Advanced Math : (Adv_Math) Questions
Question:
The ordered pair \(\mathrm{(x, y)}\) satisfies the system of equations \(\mathrm{y = x^2 + 4x + 12}\) and \(\mathrm{2x + 3 = 15}\).
What is the value of \(\mathrm{x + y}\)?
1. INFER the solution strategy
Looking at this system of equations, we have:
- \(\mathrm{y = x^2 + 4x + 12}\) (quadratic equation)
- \(\mathrm{2x + 3 = 15}\) (linear equation)
The key insight: Since the linear equation only contains x, solve it first to find x, then substitute into the quadratic equation to find y.
2. SIMPLIFY the linear equation to find x
Starting with: \(\mathrm{2x + 3 = 15}\)
- Subtract 3 from both sides: \(\mathrm{2x = 12}\)
- Divide by 2: \(\mathrm{x = 6}\)
3. SIMPLIFY by substituting x = 6 into the quadratic equation
\(\mathrm{y = x^2 + 4x + 12}\)
\(\mathrm{y = (6)^2 + 4(6) + 12}\)
Following order of operations:
- \(\mathrm{6^2 = 36}\)
- \(\mathrm{4(6) = 24}\)
- \(\mathrm{y = 36 + 24 + 12 = 72}\)
4. SIMPLIFY to find the final answer
\(\mathrm{x + y = 6 + 72 = 78}\)
Answer: 78
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make arithmetic errors when evaluating the quadratic expression.
For example, they might calculate \(\mathrm{6^2}\) as 12 instead of 36, or make addition errors when computing \(\mathrm{36 + 24 + 12}\). A common mistake is getting \(\mathrm{y = 60}\) instead of \(\mathrm{y = 72}\), which would lead to \(\mathrm{x + y = 66}\). This causes confusion since there's no clear wrong answer choice to gravitate toward, leading to guessing.
Second Most Common Error:
Poor INFER reasoning about what the question asks: Students find \(\mathrm{x = 6}\) and \(\mathrm{y = 72}\) correctly, but then give \(\mathrm{y = 72}\) as their final answer instead of \(\mathrm{x + y = 78}\).
They lose track of what the problem is actually asking for and stop one step too early. This leads to submitting 72 as their answer.
The Bottom Line:
This problem tests whether students can systematically work through a multi-step process while maintaining accuracy in their calculations and staying focused on what the question actually asks for.