Write down a system of three equations in three unknowns with exactly three solutions

Question

Write down a system of three equations in three unknowns with exactly three solutions.

I have absolutely no idea how to go about answering this question. Can someone please explain it to me?

Answer 1

Consider the following system (assuming not every equation needs to have all of the three variables in it): $$ (x-1)(x-2)(x-3)=0 \\ y=0 \\ z=0 $$ This system has exactly three solutions: $(1,0,0),(2,0,0),(3,0,0)$.

Answer 2

The first thing you should realize is that it cannot be a linear system of equations, because a linear system has either one unique solution, no solution, or infinitely many solutions.

Answer 3

If someone said "two equations, two solutions", I'd probably draw a circle and a line cutting through it. Say, $x^2+y^2 = 1$ and $x=0$. Or maybe a parabola with a line cutting it, say, $y=x^2$ and $y=1$. So my advice is "think geometrically."