Prove using an example that there is no plane on $\mathbb{R}^3$ that contains every group of 4 points

Question

Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere):

The exercise is as follows: i) Find the equation of the plane of $\mathbb{R}^3$ that contains the following three points: (0,0,1),(-1,3,5) and (1,-1,0) ii) Prove using an example that there is no plane on $\mathbb{R}^3$ that contains every group of 4 points

So I have managed to solve (i) using a lot of Google and a third party tool (my notes are not all that good), but I cannot find (ii) anywhere. Can someone prove this providing an example and explaining it properly so that I can understand the reason this is impossible?

Thanks in advance!

P.S: This is an exercise from a course on Matrixes and how to solve Systems using matrices, in case anyone wants to know.

Answer 1

From the phrasing of the question, it sounds like you understand that there is only one plane through those three points in part i). Knowing that, then we can say this:

Hint: If you find a point $P$ that does not satisfy the equation you found in i), then that point, along with the original three points, is an example of a set of four points that doesn't fit in a single plane.

Answer 2

Let $ax+by+cz=d$ be the equation of a plane (where $\{a,b,c\}\not=\{0\})$. Show that $(0,0,0),(0,0,1),(0,1,0),(0,0,1)$ cant satisfy the equation $ax+by+cz=d$ simultaneously.