Solution

Given equations are:

$x+y+z=1$ .....$(1)$

$4x+9y+16z=25$ .....$(2)$

$16x+81y+256z=625$ .....$(3)$

Now subtract $4\times(1)$ from $(2)$:

$4x+9y+16z-4x-4y-4z=25-4$

$5y+12z=21$ .....$(4)$

Now subtract $16\times(1)$ from $(3)$:

$16x+81y+256z-16x-16y-16z=625-16$

$65y+240z=609$ .....$(5)$

Multiply equation $(4)$ by $13$:

$65y+156z=273$ .....$(6)$

Now subtract $(6)$ from $(5)$:

$65y+240z-(65y+156z)=609-273$

$84z=336$

$z=4$

Put $z=4$ in equation $(4)$:

$5y+12(4)=21$

$5y+48=21$

$5y=-27$

$y=-\frac{27}{5}$

Now use equation $(1)$:

$x+y+z=1$

$x-\frac{27}{5}+4=1$

$x-\frac{27}{5}=-3$

$x=-3+\frac{27}{5}$

$x=\frac{-15+27}{5}$

$x=\frac{12}{5}$

So,

$x=\frac{36}{15}$