How to solve the system of equation with three variables

System of equation with three variables

Find the value of x⁻¹ + y⁻¹ + z⁻¹ when the system of equation with three variables are, x + y + z = 7, x² + y² + z² = 9 and x y z = 5

Solution

We know

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz

⇒ 7² = 9 + 2(xy + yz + xz)

⇒ 40 = 2(xy + yz + xz)

so, xy + yz + xz = 20

divided with x y z then

(xy + yz + xz) / (x y z) = 20/5

⇒ z⁻¹ + x⁻¹ + y⁻¹ = 4

Then the value of x⁻¹ + y⁻¹ + z⁻¹ is 4