Given the equation:
$(2,-14,8)x+(-3,21,-12)y=\vec 0$
Find the scalars x and y, not both zero, such that the above equation holds.
My attempt:
I tried solving the equation system $2x-3y=0, -14x+21y=0, 8x-12y=0$ but I only obtained trivial solutions, i.e. $x=0, y=0$
Thanks for your help.
Your effort makes a lot of sense! But you not only find the trivial solution: your system of equations is correct and it is equivalent to the only equation: $$ 2x−3y=0, $$ whose solution is $$ y=\frac{2}{3}x. $$ So, all the ordered couples of the kind $(x,y)=(x,\frac{2}{3}x)$ are solutions.
The trivial solution $(0,0)$ is an example, other ones are $(1,\frac{2}{3})$ and $(3,2)$ or $(-3,-2)$ but one linear equation in two variables has an infinity of such ordered couples of values as solutions.