Question

If $a^{x-1}=bc,b^{y-1}=ac,c^{z-1}=ab$ such that $x,y,z$ are integers then value of $xy+yz+zx–xyz$ is

Answer 1

$a^{x-1}=bc$

$\Rightarrow \frac{a^x}{a}=bc$

$\Rightarrow a^x=abc$

$\Rightarrow a=(abc{)}^{\frac{1}{x}}$---------(1)

$b^{y-1}=ac$

$\Rightarrow \frac{b^y}{b}=ac$

$\Rightarrow b^y=abc$

$\Rightarrow b=(abc{)}^{\frac{1}{y}}$-----(2)

$c^{z-1}=ab$

$\Rightarrow \frac{c^z}{c}=ab$

$\Rightarrow c^y=abc$

$\Rightarrow c=(abc{)}^{\frac{1}{z}}$-------(3)

Now, from equations (1), (2) and (3),

$abc=\left(abc{)}^{\frac{1}{x}}\right(abc{)}^{\frac{1}{y}}(abc{)}^{\frac{1}{z}}$

$\Rightarrow (abc{)}^1=(abc{)}^{\frac{1}{x}+\frac{1}{y}+\frac{1}{y}}$

$\Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{y}=1$

$\Rightarrow \frac{xy+yz+zx}{xyz}=1$

$\Rightarrow xy+yz+zx=xyz$

$\therefore xy+yz+zx-xyz=0$