What do trivial and non-trivial solution of homogeneous equations mean in matrices? [closed]

Suppose I have system of 3 equations $$a_1x+b_1y+c_1z=0$$ $$a_2x+b_2y+c_2z=0$$ $$a_3x+b_3y+c_3z=0$$ and cofficient matrix $A=\begin{equation} \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix} \end{equation}$ So I have been told that solution of this matrix will be non-trivial if $|A|=0$ and trivial in any other case. As far as I know non trivial solution means solutions is not equal to zero but in any case $x,y,z=0$ will satisfy given equations regardless of it's value of determinant. So, why do we call it "non-trivial" solution?

If $x=y=z=0$ then trivial solution And if $|A|=0$ then non trivial solution that is the determinant of the coefficients of $x,y,z$ must be equal to zero for the existence of non trivial solution. Simply if we look upon this from mathwords.com

For example, the equation $x + 5y = 0$ has the trivial solution $x = 0, y = 0.$ Nontrivial solutions include $x = 5, y = –1$ and $x = –2, y = 0.4.$