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# If we are given $x$ such that $25^x-9^y=18$ and $5^x-3^y=3$, compute $5^x+3^y$.

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If we are given $$x$$ such that $$25^x-9^y=18$$ and $$5^x-3^y=3$$, compute $$5^x+3^y$$.

Thank you so much to make the time!

SZhang

Mar 10, 2021

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I have been playing around with these two equations, and I think I have finally cracked it!

$$\fbox{1} \; 25^x-9^y = 18\\ \fbox{2} \: 5^x-3^y = 3$$

Using exponent rules, it is possible to rewrite the first equation into something that has a more obvious relationship to the second equation.

$$\fbox{1} \; 25^x - 9^y = 18\\ \left(5^2\right) ^x - \left(3^2\right)^y = 18\\ \left(5^x\right)^2 - \left(3^y\right)^2 = 18$$

Through some algebraic manipulation, the relationship between equation 1 and 2 is clearer. Let's use that fact to our advantage. I will multiply both sides of  equation 2 by $$5^x + 3^y$$. This allows for some simplification, and the answer falls out in the end.

$$\fbox{2} \; 5^x - 3^y = 3\\ (5^x - 3^y)(5^x + 3^y) = 3(5^x + 3^y)\\ \left(5^x\right)^2 - \left(3^y\right)^2 = 3(5^x + 3^y)\\ 18 = 3(5^x + 3^y)\\ 5^x + 3^y = 6$$

Mar 10, 2021
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You did it! It was right! Good job!

Mar 10, 2021
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