Question

Find k for which the set of equations
$x+y-2z=0,2x-3y+z=0,x-5y+4z=k$
are consistent and find the solutions for all such values of k.

Answer 1

Here
$△=\left|\begin{array}{ccc}1& 1& -2\\ 1& -3& 1\\ 1& -5& 4\end{array}\right|=1(-12+5)-1(6-5)-2(-10+3)=-7-7+14=0$
Thus $△=0$ hence if the equations are to be consistent, then we must have
${△}_x={△}_y={△}_z=0$
${△}_x=\left|\begin{array}{ccc}0& 1& -2\\ 0& -3& 1\\ k& -5& 4\end{array}\right|=0$
if $5k=0\therefore k=0$