Creating a System of Equations given the roots of the equation

Question

Say I was given three roots of a $3$ variable system, where $x=3, y=2, z=-8$.

I was then instructed to create a three equation / three variable system of equations from those roots (edited).

How can that be done?

Answer 1

The comment of Ross Millikan gives a very simple and direct answer, if you want a more interesting system of equations just put any random coefficients to $x$, $y$ and $z$:

$3x-y+2z = a$

$5x-y-3z = b$

$x+y+z = c$

and calculate $a$, $b$ and $c$ replacing $x=2$, $y=3$ and $z=-8$.

This will give you a system that has that solution BUT it may have more solutions, depending of the coefficients one choose.