What does it mean for a system of equations to have a non trivial solution?

Trivial means obviously but how can a equation have a obvious solution?

My book says if solution is non trivial than determinant of coefficient of variables is 0.

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The word is often used about systems of equations such as $$ 3x+5y-12z = 0 \\ x-y+5z = 0 \\ 5x+y+3z=0 $$ where it is immediately obvious that setting all of the unknowns to $0$ will solve the system. The question is then whether the system has other solutions than that.

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    $\begingroup$ Well, sometimes the system is displaced e.g. $$3(x-2)+5y-12z = 0 \\ (x-2)-y+5z = 0 \\ 5(x-2)+y+3z=0$$ gives the trivial solution $x=2,y=0$ and $z=0$. Btw I think your have a nice answer. $\endgroup$ – manooooh May 8 '19 at 16:59

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