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Prove that every first degree equation in $x$ and $y$ always represents a straight line.

My Attempt: Let the first degree equation in $x$ and $y$ be $$ax+by+c=0$$. Let $A(x_1, y_1)$, $B(x_2, y_2)$ and $C(x_3,y_3)$ be any three points in the locus of the given first degree equation. Then, $$ax_1+by_1+c=0$$ $$ax_2+by_2+c=0$$ $$ax_3+by_3+c=0$$

Now, how should I complete?

pi-π
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    How do you define "a straight line"? – 5xum Jul 31 '17 at 13:26
  • @5xum, The collection of collinear points that has no end. – pi-π Jul 31 '17 at 13:29
  • Unfortunately your definition is circular. "Collinear" means "lying on a straight line", hence you have defined "a straight line" to be "a collection of points lying on a straight line that has no end". – Lee Mosher Jul 31 '17 at 13:31
  • @blue_eyed_... How do you define "colinear"? – 5xum Jul 31 '17 at 13:34
  • @5xum, lying on the same straight line. – pi-π Jul 31 '17 at 13:35
  • @blue_eyed_... I hope you are joking. – 5xum Jul 31 '17 at 13:36
  • if you take two Points say $$A(x_1,y_1)$$ and $$B(x_2,y_2)$$ and $$A\neq B$$ you can compute the Parameters $a,b,c$ – Dr. Sonnhard Graubner Jul 31 '17 at 13:36
  • @5xum, For what purpose? – pi-π Jul 31 '17 at 13:39
  • @blue_eyed_... You define "being colinear" as "lying on the same straight line". And you define "straight line" as "a collection of colinear points". This means that a "straight line" is "a collection of points lying on the same straight line". The definitions are circular, and therefore useless. – 5xum Jul 31 '17 at 13:40
  • @5xum, I know those things. Please say me the correct ones. – pi-π Jul 31 '17 at 13:43
  • @blue_eyed_... To me, $ax+by+c=0$ is the definition of a straight line. – 5xum Jul 31 '17 at 13:43
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    Without knowing where you got this problem from, we cannot tell you "the correct ones". That's why we are asking you. Did this problem come from a book, and if so how does that book define a straight line? Or, where else did this problem come from, and how is a straight line defined in that source? – Lee Mosher Jul 31 '17 at 14:14

1 Answers1

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Hint:

Show the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are linearly dependent.

Bernard
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