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Question 26

If $$\frac{4^x}{2^{x+y}} = 8$$ and $$\frac{9^{x+y}}{3^y} = 243$$, then $$x^2 + xy + 7y^2 =$$


Correct Answer: 7

Solution

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The first equation gives $$2^{2x - x - y} = 2^3$$, so $$x - y = 3$$, and the second gives $$3^{2x + 2y - y} = 3^5$$, so $$2x + y = 5$$. Adding, $$3x = 8$$, so $$x = \frac{8}{3}$$ and $$y = -\frac{1}{3}$$. Then $$x^2 + xy + 7y^2 = \frac{64}{9} - \frac{8}{9} + \frac{7}{9} = \frac{63}{9} = 7$$.

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