How do you solve a system of equation with a variable as an exponent? So, I've been doing some random math on my own and I ran into a problem. I was met with the system of equations of $$x=3^y$$ $$x+1=5^y$$ How can I solve this algebraically without using the graphs?

Eliminating $x$ from the first equation and replacing in the second let you with the equation $$3^y+1=5^y$$ which is highly nonlinear (and you look for the intersection of these two functions).
You can make the problem nicer taking logarithms and get $$y\log(5)=\log(3^y+1)$$ which is now the intersection of a curve and a straight line. 
For solving the equation $$f(y)=y\log(5)-\log(3^y+1)=0$$ let us use Newton method which, starting from a "reasonable" guess $y_0$, will update it according to $$y_{n+1}=y_n-\frac{f(y_n)}{f'(y_n)}$$ $$f'(y)=\log (5)-\frac{3^y \log (3)}{3^y+1}$$ By inspection, you know that the solution for $y$ is between $0$ and $1$  since $f(0)=-\log (2)<0$ and $f(1)=\log \left(\frac{5}{4}\right)>0$. So, let us start iterating at $y_0=1$; the successive iterates of Newton method are then $0.7159139357$, $0.7271409790$, $0.7271601514$ which is the solution for ten significant figures.
Now, compute $x$ from $x=3^y$ and get $x=2.223020992$
