Question

Let the intersection of the three planes
$2x+y-4z-17=0.$
$3x+2y-2z-25=0$
and $2x-4y+3z+25=0$ be at a point $(k,m,n).$ Find $k+m+n$ ?

Answer 1

$2x+y-4z-17=0$ ..... $\left(i\right)$
$3x+2y-2z-25=0$ ..... $\left(ii\right)$
$2x-4y+3z+25=0$ ...... $\left(iii\right)$
$\left(i\right)-\left(iii\right)$

$\Rightarrow 5y-7z=42$ ...... $\left(iv\right)$
$2\left(ii\right)-3\left(i\right)$

$\Rightarrow y+8z=-1$ ..... $\left(v\right)$
Solving $\left(iv\right)$ and $\left(v\right)$ we get $y=7,z=-1$ and by $\left(i\right),$ $x=3$
Thus, $(k,m,n)=(3,7,-1)$

Hence, $k+m+n=3+7-1=9$.