Take the equations $$x+y=5$$ $$x^2 + y^2 =13$$
The most basic method to solve this system is to first express the linear equation in terms of one of the variables and then sub that into the non-linear equation.
But I am curious if there are other methods to solve such a system ?
Best Answer
In general, the set of equations:
$$\sum_{k=1}^{N}x_k^p = S_p$$
for $1\leq p\leq N$, can be solved by considering the function:
$$f(x) = -\sum_{p=1}^N\log\left(1-\frac{x_p}{x}\right) \tag{1}$$
The expansion of $f(x)$ around infinity is given by:
$$f(x) = \sum_{r=1}^{\infty}\frac{S_r}{r x^r}$$
We can thus write down $f(x)$ to order $x^{-2}$ as:
$$f(x) = \frac{5}{x} + \frac{13}{2 x^2} + \mathcal{O}\left(x^{-3}\right)\tag{2}$$
From (1) it follows that $x^2 \exp\left[-f(x)\right]$ is a second degree polynomial that has the solutions as its roots. Using (2) it follows that:
$$\exp\left[-f(x)\right] = 1 - \frac{5}{x} + \frac{6}{x^2} + \mathcal{O}\left(x^{-3}\right)$$
It thus follows that:
$$(x-x_1)(x-x_2) = x^2 - 5 x + 6$$
So, the solutions are $x_1=2$ and $x_2 = 3$ and vice versa.