Question

Find the value of $x,y$ and $z$ from the given linear equations: $x+y+z=0,2x+5y+7z=0,2x-5y+3z=0$

Answer 1

Let,

$x+y+z=0$....(1)

$2x+5y+7z=0$....(2)

$2x-5y+3z=0$....(3)

Adding equation (2) and (3), we get,
$2x+5y+7z+2x-5y+3z=0$
$4x+10z=0$
$\Rightarrow 4x=-10z$
$\therefore x=\frac{-10z}{4}$....(4)

Subtracting (3) from (2), we get,
$2x+5y+7z-2x+5y-3z=0$
$10y+4z=0$
$10y=-4z$
$\therefore y=\frac{-4z}{10}$....(5)

Substituting the values of $x$ and $y$ in equation (1), we get,

$\Rightarrow \frac{-10z}{4}+\frac{-4z}{10}+z=0$

$\Rightarrow (\frac{-10}{4}+\frac{-4}{10}+1)z=0$
$\Rightarrow z=\frac{0}{(\frac{-10}{4}+\frac{-4}{10}+1)}$
$\therefore z=0$

Substituting the value of $z$ in equations (4) and (5), we get,

$\Rightarrow x=0,y=0$ as multiplying $0$ with any non zero quantity gives us zero only.

Hence, $x=y=z=0$