<s>[INST] Exponential Simultaneous Equations Solve the following simultaneous equations:
$$2^x + 2^y = 10$$
$$x + y = 4$$
Looking at it, it is obvious that the answers are $(3,1)$ and $(1,3)$, however, I was wondering if they could be solved algebraically. Here's my approach:
$$2^x + 2^{4-x} = 10$$
$$2^x + \frac{(2^4)}{(2^x)} = 10$$
$$2^x + \frac {16}{2^x} = 10$$
And this is where I get stuck. Any help will be greatly appreciated, thanks in advance.
 [/INST] Alternatively,
$$2^x+2^y=10\\2^x\cdot2^y=16.$$
You know the sum and the product, so by $\left(2^x-2^y\right)^2=\left(2^x+2^y\right)^2-4\cdot 2^x\cdot 2^y$, you get
$$2^x,2^y=2,8.$$
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