This is a follow-up to this question: Proving that two systems of linear equations are equivalent if they have the same solutions
Prove that if two homogenous systems of linear equations in two unknowns have the same solutions, then they are equivalent.
I found this problem in Hoffman and Kunze's Linear Algebra book. Since row operations or anything like that haven't yet been introduced at this point, I want to solve this without using row operations.
For that reason, this question is not really a duplicate because I'm asking for a different way to solve the problem- something along the lines of the 2nd answer in the linked question, but I have a doubt about that.
Comparing this equation with all the equations of the second system and also utilising the fact given that both the systems have the same solutions
Why? We can only equate coefficients of equations in variables $x_1, x_2$ if they're the same for all values of $x_1, x_2$, and not just specific values for them. Secondly, since we're free to select the scalars $C_1, C_2, \ldots$, that indicates any equation in the second system can be made an arbitrary linear combination of the equations in the first system.
I'm probably missing something obvious but it would be nice to get some clarification on a non-row operation method.