Question

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

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (D)

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 $3x-y-z=0$
Now, $\Delta =\left|\begin{array}{ccc}1& 1& -1\\ 3& -1& -1\\ 1& -3& 1\end{array}\right|$
$=1(-1-3)-1(3+1)-1(-9+1)$
$=-4-4+8=0$
$\therefore$ It has infinite solution.
Note: It $\Delta \ne 0$, then the only solution is $x=y=z=0$.