The number of solutions of the system of the system

Question:

The number of solutions of the system of the system of equations $x+2 y+z=3,2 x+3 y+z=3,3 x+5 y+2 z=1$ is___________

Solution:

The given system of equations is $x+2 y+z=3,2 x+3 y+z=3$ and $3 x+5 y+2 z=1$.

$\Delta=\left|\begin{array}{lll}1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 5 & 2\end{array}\right|=1(6-5)-2(4-3)+1(4-3)=1-2+1=0$

$\Delta_{x}=\left|\begin{array}{lll}3 & 2 & 1 \\ 3 & 3 & 1 \\ 1 & 5 & 2\end{array}\right|=3(6-5)-2(6-1)+1(15-3)=3-10+12=5 \neq 0$

Here, $\Delta=0$ and at least one of $\Delta_{x}, \Delta_{y}$ and $\Delta_{z}$ is not zero, so the given system of equations has no solution.

Thus, the given system of equations has no solution.

The number of solutions of the system of the system of equations x + 2y + z = 3, 2x + 3y + z = 3, 3x + 5y + 2z = 1 is ___0___

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