If x - 2y = 2 and x ^ 2 + 4xy + 4y ^ 2 = 8, what is the value of xy?

Rationale

xy equals 1/2.

To find the value of xy, we can manipulate the given equations. Starting from the first equation, x - 2y = 2, we can express x in terms of y, and then substitute into the second equation to solve for xy.

A (01-Feb) CORRECT

By substituting x = 2 + 2y into the second equation, we derive that (2 + 2y)^2 + 4(2 + 2y)(y) + 4y^2 = 8 simplifies to yield xy = 1/2. This value satisfies both original equations, confirming it as the solution.

B (1) YOUR ANSWER

If we assume xy = 1, substituting back into the second equation would not satisfy the equation since the derived expressions do not hold true. Therefore, this choice does not match the values obtained from the equations.

C (2) YOUR ANSWER

Assuming xy = 2 leads to a contradiction when substituted back into the original equations, revealing that the values of x and y derived from this assumption would not satisfy the necessary conditions set by the equations.

D (02-May) YOUR ANSWER

If we consider xy = 5/2, substituting this value into either of the equations results in a disparity that cannot be resolved, hence failing to satisfy the original equations. Thus, this choice is not valid.

Conclusion

The correct value of xy, derived through manipulation of the equations provided, is 1/2. This value satisfies both equations and maintains the relationships established in the problem. Therefore, it accurately reflects the solution sought in the question.

If x - 2y = 2 and x ^ 2 + 4xy + 4y ^ 2 = 8, what is the value of xy?

xy equals 1/2.

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