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## Homework Statement

Suppose (a,b) and (c,d) are unique points in ##\mathbb{R}^{2}## that satisfy Ax+By+C=0. Suppose also that the solutions of Ax+By+C=0 form a line. Then the line that (a,b) and (c,d) lie on is unique.

## Homework Equations

I showed in a previous part of the problem that there exists real numbers A,B,C not all equal to 0 that satisfy Aa+Bb+C=0 and Ac+Bd+C=0. I also showed that if A',B',C' were also nontrivial solutions that satisfied the two equations then A',B',C' were scalar multiples of A,B,C.

## The Attempt at a Solution

Suppose that (a,b) and (c,d) are unique points that satisfy Ax+By+C=0. Suppose also that the set of solutions of Ax+By+C=0 forms a line that is not unique so there exists another equation such that A'x+B'y+C'=0 but we previously showed that A',B',C' must be scalar multiples of A,B,C so the two lines are actually the same. Hence the line that (a,b) and (c,d) lie on is unique.

I thought that since A',B',C' were scalar multiples of A,B,C then the new equation was just a stretched version of the older line so they must be the same line. Is this logic correct?