Solve the equation with the help of determinants

Question

The following set of equations is given:

$$\begin{cases}{{x^2z^3\over y}=e^8\\{y^2z\over x}=e^4\\{x^2y\over z^4}=1}\end{cases}$$

I have solved problems before with three variables in three equations using determinant method, that is Cramer's rule. But in this problem the variables are in multiplication. I seem to have no idea on how to solve this. Please help ASAP.

Answer 1

Multiplying all equations together we get $$e^{12}=x^4y^2$$ so $$y=\pm\frac{e^6}{x^2}$$ plugging this in the second equation we get $$z=\frac{x^5}{e^8}$$ plugging this in the third equation we get $$x^4=e^{-6}$$ Can you finish?

Answer 2

Hint

Take $\log$ from both sides of each of the equations above and define $$a=\log x\\b=\log y\\c=\log z$$then you are able to use Cramer's rule again for $a,b,c$ and then find $x,y,z$.