$\begin{gathered} \frac{{{x_1} + 2{x_2} + {2^2}{x_3} + {2^3}{x_4} + {2^4}{x_5}}}{{2\left( {{x_6} + 2{x_7} + {2^2}{x_8} + {2^3}{x_9} + {2^4}} \right)}} = \frac{{1034}}{{1625}} \hfill \\ \frac{{{x_1} + 3{x_2} + {3^2}{x_3} + {3^3}{x_4} + {3^4}{x_5}}}{{3\left( {{x_6} + 3{x_7} + {3^2}{x_8} + {3^3}{x_9} + {3^4}} \right)}} = \frac{{1975}}{{4522}} \hfill \\ \frac{{{x_1} + 4{x_2} + {4^2}{x_3} + {4^3}{x_4} + {4^4}{x_5}}}{{4\left( {{x_6} + 4{x_7} + {4^2}{x_8} + {4^3}{x_9} + {4^4}} \right)}} = \frac{{323}}{{966}} \hfill \\ \frac{{{x_1} + 5{x_2} + {5^2}{x_3} + {5^3}{x_4} + {5^4}{x_5}}}{{5\left( {{x_6} + 5{x_7} + {5^2}{x_8} + {5^3}{x_9} + {5^4}} \right)}} = \frac{{26887}}{{99000}} \hfill \\ \frac{{{x_1} + 6{x_2} + {6^2}{x_3} + {6^3}{x_4} + {6^4}{x_5}}}{{6\left( {{x_6} + 6{x_7} + {6^2}{x_8} + {6^3}{x_9} + {6^4}} \right)}} = \frac{{2676}}{{11687}} \hfill \\ \end{gathered} $
This Diophantine equation group is derived from the Com Quantum Law fitting of the GW150914 signal of LIGO and it has a set of solutions called the least rational solution,
$\begin{gathered} {x_1} = \frac{{63}}{{16}},\quad {x_2} = \frac{{447}}{{32}},\quad {x_3} = \frac{{69}}{4},\quad \hfill \\ {x_4} = \frac{{69}}{8},\quad {x_5} = \frac{3}{2},\quad {x_6} = \frac{{105}}{{32}},\quad \hfill \\ {x_7} = \frac{{389}}{{32}},\quad {x_8} = \frac{{227}}{{16}},\quad {x_9} = \frac{{13}}{2} \hfill \\ \end{gathered} $
This is a group solution found by computer. So do the equations have other rational solutions? Different solutions correspond to the same Com Quantum Law. It may take several months for experts in the field of non number theory and computer science to find the above set of solutions. So, is there a general solution method for such Diophantine equations?
You can reduce each of your five equations to a linear equation so you get a system of five linear equations with nine unknowns $$a_{11}x_1+\cdots a_{19}x_9=c_1\\a_{21}x_1+\cdots a_{29}x_9=c_2\\.......\\a_{51}x_1+\cdots+a_{59}x_9=c_5$$ and this have infinitely many solutions. In fact, you can take arbitrary values, for example, for $x_6,x_7,x_8$ and $x_9$ so you get a system of five equations with five unknowns $$a_{11}x_1+\cdots a_{15}x_5=A_1\\a_{21}x_1+\cdots a_{25}x_5=A_2\\.......\\a_{51}x_1+\cdots+a_{55}x_5=A_5$$
and for each of such a systems with determinant distinct of zero you have a rational solution for each value of the resultant $A_i$. If all the determinants (there are $\binom95=126$ in total) are null also you have infinitely many solutions.
Example: Make, for example, $x_6=x_7=x_8=x_9=1$ and you will have the system $$\begin{pmatrix}a_1&2a_1&2^2a_1&2^3a_1&2^4a_1\\a_2&3a_2&3^2a_2&3^3a_2&3^4a_2\\a_3&4a_3&4^2a_3&4^3a_3&4^4a_3\\a_4&5a_4&5^2a_4&5^3a_4&5^4a_4\\a_5&6a_5&6^2a_5&6^3a_5&6^4a_5\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\\x_4\\x_5\end{pmatrix}=\begin{pmatrix}A_1\\A_2\\A_3\\A_4\\A_5\end{pmatrix}$$ where $A_1=2a_1(2^5-1), A_2=\dfrac{3a_2(3^5-1)}{2},A_3=\dfrac{4a_3(4^5-1)}{3},A_4=\dfrac{5a_4(5^5-1)}{4}$ and $A_5=\dfrac{6a_5(6^5-1)}{5}$
If the determinant is non-zero there is a unique solution according when $x_6,x_7,x_8$ and $x_9$ are all equal to $1$ as we have chosen but you can vary these arbitrarily and you get infinitely many solutions.
And if the determinant is null, a fortiori you have infinitely many solutions, for example the system $$2x+3y=1\\4x+6y=2$$ has determinant zero but it is the equation of a rightline $2x+3y=1$ which have an infinity of points as you know.