I was listening to a lecture video my algebra teacher sent us, about systems of equations. As I understood, we can show our system of equations as lines with points at which they cross, where each line presents an equation and and the crossing points present solutions. In pdf it says that, for example:
For a system of equations with 3 equations and 2 unknowns, we can either have 1 point(solution) where they intersect, or infinitely many solutions if they are all overlapping. My question is:
Can't we have 3 solutions for example? Or any other amount more than 1 but not infinite?
Best Answer
Suppose the number of solutions is more than $1$ but less than infinity. Then it is at least $2$. Find $2$ points that are solutions to all 3 equations. Then every equation is a straight line going through these two points. Since there is one unique line going through two distinct points, this means that each of the $3$ equations is the same line. But this means that there are infinitely many points that solve all $3$ equations simultaneously. This contradicts the assumption that “the number of solutions is more than $1$ but less than infinity”.