Question: Prove the point $(2,5,-4)$ is a solution to the two equations:
$$ x + 2y + 3z = 0 $$ $$ 2x-y-2z=7$$ Find any other equation in the form $$ax + by + cz = k $$ that passes through $(2,5,-4)$ and demonstrate that it gives a unique solution with the existing two equations.
For the first part I "proved" that $(2,5,-4)$ is a solution by substitution and showing the LHS=RHS hence the set is a solution.. However I am stuck trying to find an equation in the form $$ax + by + cz = k $$ that passes through $(2,5,-4)$ algebraically..
$k=2a+5b-4c$ with any $a,b,c$ such that $\left| \begin{array}{ccc} 1 & 2 & 3 \\ 2 & -1 & -2 \\ a & b & c \end{array} \right| \neq 0$ will do.
For simplicity, may choose $a=1$, $b=c=0$, $k=2$.