Given that $$x, y, z$$ are positive real numbers, if $$(x + y)^2 - z^2 = 8, (y + z)^2 - x^2 = 10  and  (x + z)^2 - y^2 = 7,  then  (x + y + z)$$ is equal to:

Given $$\left(x+y\right)^2-z^2=8$$ ................(1)

$$\left(y+z\right)^2-x^2=10$$...............(2)

$$\left(x+z\right)^2-y^2=7$$ ................(3)

Adding equations (1),(2),(3) we get

$$\left(x+y\right)^2-z^2+\left(y+z\right)^2-x^2+\left(x+z\right)^2-y^2=8+10+7$$

$$x^2+y^2+2xy-z^2+y^2+z^2+2yz-x^2+x^2+z^2+2zx-y^2=25$$

$$x^2+y^2\ +\ z^2+2xy+2yz+2zx=25$$

$$\left(x+y+z\right)^2=25$$

$$=$$>    $$x+y+z=5$$