Can you use Gaussian Elimination for three non linear equations?
The equations are
$na^{n-1}b=12$
$\frac{n\left(n-1\right)}{2}a^{n-2}b^2=60$ and
$\frac{n\left(n-1\right)\left(n-2\right)}{6}a^{n-3}b^3=160$
I know I could brute force it easily by eliminating one variable at a time, but perhaps there is a methodic way for three systems of equations containing exponential functions as well.
$\endgroup$ 31 Answer
$\begingroup$If you want something else than brute force, divide $(2)$ by $(1)$ and $(3)$ by $(2)$ to obtain$$\frac{b (n-1)}{2 a}=5 \tag 4$$$$\frac{b (n-2)}{3 a}=\frac{8}{3}\tag 5$$ Divide $(5)$ by $(4)$$$\frac{2 (n-2)}{3 (n-1)}=\frac{8}{15} \implies n=6$$ Go back to $(4)$ and $(5)$ to get $a$ and $b$.
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