+41 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
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.
\(\fbox{2} \; 5^x - 3^y = 3\\ (5^x - 3^y)(5^x + 3^y) = 3(5^x + 3^y)\\ \left(5^x\right)^2 - \left(3^y\right)^2 = 3(5^x + 3^y)\\ 18 = 3(5^x + 3^y)\\ 5^x + 3^y = 6\)
