Difficulty: Medium
Correct Answer: -1
Explanation:
Introduction / Context:
This question combines simultaneous equations and algebraic identities to find the product xy without necessarily solving for x and y individually. It is a neat example of how squaring a linear equation and comparing with a quadratic relationship can reveal cross terms like xy.
Given Data / Assumptions:
Concept / Approach:
Main ideas:
Step-by-Step Solution:
Start from 3x + 4y = 6.
Square both sides: (3x + 4y)^2 = 6^2.
Expand the left side: 9x^2 + 24xy + 16y^2 = 36.
We are also given 9x^2 + 16y^2 = 60.
Subtract the quadratic equation from the squared equation:
(9x^2 + 24xy + 16y^2) − (9x^2 + 16y^2) = 36 − 60.
This simplifies to 24xy = −24.
Therefore xy = −24 / 24 = −1.
Verification / Alternative check:
We can solve the system explicitly to verify. From 3x + 4y = 6, express x in terms of y and substitute into 9x^2 + 16y^2 = 60. Solving this pair will give numerical values of x and y whose product equals −1, confirming the derived result without needing to show every intermediate step here.
Why Other Options Are Wrong:
Option b: 1 would require 24xy = 24, contradicting the calculated equation 24xy = −24.
Option c and option d: −2 or 2 would require different constants on the right side of the equation obtained after subtraction.
Option e: 0 would mean 3x + 4y squared equals 9x^2 + 16y^2, which happens only if xy = 0, inconsistent with 24xy = −24.
Common Pitfalls:
Students may incorrectly expand (3x + 4y)^2 or mis-handle the subtraction of the two quadratic equations, leading to sign mistakes. Another pitfall is trying to solve directly for x and y when a more elegant approach via comparison is faster and less error prone.
Final Answer:
The value of the product xy is −1.
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