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  4. The number of solutions of equations x+y-z=0,3 x-y-z=0,x-3 y+1=0 is:

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Q. The number of solutions of equations $ x+y-z=0,3\,x-y-z=0,x-3\,y+1=0 $ is:

Bihar CECEBihar CECE 2001

Solution:

If system of equations is a homogeneous equation and its determinant is zero, then the solution of the given equation is infinitely. Given system of equations are
$x+y-z=0$, and $3 \,x-y-z=0$
Now, $\Delta=\begin{vmatrix}1 & 1 & -1 \\ 3 & -1 & -1 \\ 1 & -3 & 1\end{vmatrix}$
$=1(-1-3)-1(3+1)-1(-9+1)$
$=-4-4+8=0$
$\therefore $ It has infinite solution.
Note: It $\Delta \neq 0$, then the only solution is $x=y=z=0$.