[Math] Suppose we want to express the point $(2,3)$ in $\mathbb{R}^2$ as the solution space of a linear system of equations.

Question

I'm in the middle of figuring out how to do this question:

Suppose we want to express the point $(2,3)$ in $\mathbb{R}^2$ as the solution space of a linear system of equations.

(a) What is the smallest number of equations you would need? Write down such a system.

(b) Can you add one more equation to the system in (a), in such a way that the new system still has the unique solution $(2,3)$?

(c) What is the maximum number of distinct equations you can add to your system in (a) to still maintain the unique solution $(2,3)$?

(d) Is there a general form for the equations in (c)?

I tried to figure this out. Specifically for question (c), I do understand that we are given a line. In order to have a unique solution, we need another line that can intersect the original one. But now I'm having doubts whether the maximum number of distinct equations is infinite? Because technically our solution is the intersection point, and any "random lines" that go through this point is considered a solution. Am I right?

Answer 1

Yes, for c you can add as many lines as you want that go through $(2,3)$ and still have it solve all the equations.