Proof for the statement: "the equation ax+by+c=0 always expresses only 1 line"

[Zach001]

Can I get the proof for the statement: "the equation ax+by+c=0 always expresses only 1 line"

 

[blamocur]

Can I get the proof for the statement: "the equation ax+by+c=0 always expresses only 1 line"

This depends on what you consider "proof". I.e., what is the set of axioms and/or theorems which we can take for granted ?

Would it be enough to prove that any 3 points satisfying the equation lie on a straight line? In this case we can refer to Euclid's axioms, i.e., there is exactly one straight line for any pair of points.

 

[Dr.Peterson]

Can I get the proof for the statement: "the equation ax+by+c=0 always expresses only 1 line"

But what if a=b=c=0? Was any condition mentioned? Please state the entire problem you are working on.

 

[pka]

Suppose that $\ell_1~\&~\ell_2$ are two lines represented by the general linear equation $Ax+By+C=0$
Can you explain the following? From the given there must be a point $(p,q)$. on one of lines but not the other.
Say it is is on $\ell_1$. Because $Ax+By+C=0$ determines $\ell_1$ we have $Ap+Bq+C=0$, WHY?
Now can you explain why $(p,q)\notin\ell_2\text{ means that }Ap+Bq+C\ne 0~?$
Please reply with what you are able to do.




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