How to solve this multivariable exponential equation? I searched if this was asked before but couldn't find a solution. I have this equation 

$y^{70} = x + 500 $
$y^{50} = x + 1 $

Is it possible to solve this equation? The only thing I could do is to bring it into this form and then cross-multiply which didn't yield many results. 

$y^{20} = \frac{x+500}{x+1} $


The best I can think of at the moment is
$$
\eqalign{
  & \left\{ \matrix{
  50\ln y = \ln \left( {1 + x} \right) \hfill \cr 
  20\ln y = \ln \left( {{{x + 500} \over {x + 1}}} \right) = \ln \left( {1 + {{499} \over {x + 1}}} \right) \hfill \cr}  \right.  \cr 
  & \ln y = {1 \over {50}}\ln \left( {1 + x} \right) = {1 \over {20}}\ln \left( {1 + {{499} \over {x + 1}}} \right)  \cr 
  & \left( {1 + x} \right)^{\,2/5}  = 1 + {{499} \over {x + 1}}  \cr 
  & \left( {1 + x} \right)^{\,7/5}  = x + 1 + 499  \cr 
  & 1 + x = u^{\,5}   \cr 
  & u^{\,7}  - u^{\,5}  = u^{\,5} \left( {u^{\,2}  - 1} \right) = 499 \cr} 
$$
which clearly has only one solution, and that numerically is easy to solve.
