Question

x^2=y+z,y^2=z+x,z^2=x+y then the value of 1/x+1/+1/y+1+1/z+1=

Answer 1

Step-by-step explanation:

Given : x² = y + z

$\mathsf{\implies x = \dfrac{y + z}{x}}$

Given : y² = x + z

$\mathsf{\implies y = \dfrac{x + z}{y}}$

Given : z² = y + x

$\mathsf{\implies z = \dfrac{y + x}{z}}$

$\mathsf{Now,\;Consider :\;\dfrac{1}{x + 1} + \dfrac{1}{y + 1} + \dfrac{1}{z + 1}}$

Substituting the values of x , y , z in above expression, We get :

$\mathsf{\implies \dfrac{1}{\bigg[\dfrac{y + z}{x}\bigg] + 1} + \dfrac{1}{\bigg[\dfrac{z + x}{y}\bigg] + 1} + \dfrac{1}{\bigg[\dfrac{x + y}{z}\bigg] + 1}}$

$\mathsf{\implies \dfrac{1}{\bigg[\dfrac{y + z + x}{x}\bigg]} + \dfrac{1}{\bigg[\dfrac{z + x + y}{y}\bigg]} + \dfrac{1}{\bigg[\dfrac{x + y + z}{z}\bigg]}}$

$\mathsf{\implies \dfrac{x}{x + y + z} + \dfrac{y}{x + y + z} + \dfrac{z}{x + y + z}}$

$\mathsf{\implies \dfrac{x + y + z}{x + y + z}}$

$\mathsf{\implies 1}$