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y = x\(^2\) 2y + 6 = 2(x + 3)If (x, y) is a solution of the system of equation...


Question

y = x\(^2\)
2y + 6 = 2(x + 3)

If (x, y) is a solution of the system of equations above and x > 0, what is the value of xy ?

Options

A)
1
B)
2
C)
3
D)
9

The correct answer is A.

Explanation:

Choice A is correct. Substituting x\(^2\) for y in the second equation gives 2(x\(^2\)) + 6 = 2(x + 3). This equation can be solved as follows:

2x\(^2\) + 6 = 2x + 6 (Apply the distributive property.)
2x\(^2\) + 6 − 2x − 6 = 0 (Subtract 2x and 6 from both sides of the equation.)
2x\(^2\) − 2x = 0 (Combine like terms.)
2x(x − 1) = 0 (Factor both terms on the left side of the equation by 2x.)

Thus, x = 0 and x = 1 are the solutions to the system. Since x > 0, only x = 1 needs to be considered. The value of y when x = 1 is y = x\(^2\) = 1\(^2\) = 1. Therefore, the value of xy is (1)(1) = 1.

Choices B, C, and D are incorrect and likely result from a computational or conceptual error when solving this system of equations.


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