Solving the system with $y=x$ and $ y=(1-40x)^{-1/51.33}-1$

Question

Solve for $(x,y)$: \begin{align} y &= x\\ y &= (1-40x)^{-1/51.33}-1\ \end{align}

I don't know if finding a closed-form solution is possible, have been trying for almost half an hour but still get nothing.

Please tell me some idea about how to solving it, or if it is possible to do.

Edit: I did manage to get the answer via Geogebra, but I would love to see how people solve it analytically.

Answer 1

$$y = x \land y = (1-40x)^{-1/51.33}-1\\ \implies (1-40x)^{-1/51.33}-x=1\\ \implies \frac{1}{(1 - 40 x)^{1/51.33}} = x+ 1 \qquad 40x\ne1\\ \implies 1=(x+ 1)(1 - 40 x)^{1/51.33} \\ $$ At this point I must defer to WolframAlpha here for $\quad x=y=0\land x=y = 0.00999525$

Answer 2

Using whole numbers, consider that you look for the zero of function $$f(x)=(1-40x)^{-\frac {100}{5133}}-x-1$$ It is clear that $x=0$ it a trivial solution. Discarding this one, we can approximate the other building the $[1,n]$ Padé approximant of $\frac{f(x)}x$ around $x=0$. This will write $$\frac{f(x)}x \sim \frac{ a^{(n)} x -\frac{1133}{5133}} {1+\sum _{i=1}^n b_i^{(n)} x^i }$$ where the coefficients $ a^{(n)}$ and $b_i^{(n)}$ are compued from the function and derivative values at $x=0$.

Then the approximate solution is just given by $$x_{(n)}=\frac{1133}{5133\,a^{(n)}}$$

This will generate a sequence of rational numbers, the very first ones being $$\left\{\frac{5815689}{418640000},\frac{17447067}{1725707120},\frac{24440327087}{243 9163381610},\frac{21844121555788638898671}{2184051507726409040779072}\right\}$$

Let us look at their decimal representation $$\left( \begin{array}{cc} n & x_{(n)} \\ 1 & 0.0101100974 \\ 2 & 0.0100199631 \\ 3 & 0.0100016513 \\ 4 & 0.0099970864 \\ 5 & 0.0099958134 \\ 6 & 0.0099954326 \\ 7 & 0.0099953131 \\ 8 & 0.0099952742 \\ 9 & 0.0099952613 \\ 10 & 0.0099952569 \\ 11 & 0.0099952553 \\ 12 & 0.0099952548 \\ 13 & 0.0099952546 \\ 14 & 0.0099952546 \\ 15 & 0.0099952545 \end{array} \right)$$

Just for the fun $$x_{(8)}=\frac{2076678199429529816581224448972052669538078027653922391}{20776600532441929227 8132724022546069652413472361607062796}$$