Solve to get a value for $y.$

Question

Please help me solve this question. Solve for y.

$1.\quad a^x=b^y=c^z$

$2.\quad b^2=ac$

I figured out that $b^{2+y}=a^{x+1}c^{z+1}$, but I am not able to go further, please help me.

Answer 1

Substituting $$a^x=b^y=c^z=t$$ so we get (using $b^2=ac$) $t^{2/y}=t^{1/x+1/z}$ and we get $$\frac{2}{y}=\frac{1}{x}+\frac{1}{z}$$