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There are many equations in Analytic Geometry like equation of a line, equation of a plane etc.

My question:

1) Why equations instead of functions?

2) Why do equations almost always equal zero?

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    $\begingroup$ I'm not sure if I'm parsing your question right, but: not all curves can be expressed as functions (e.g. circles), since functions are supposed to satisfy the "vertical line test", and most curves don't... $\endgroup$ – J. M. is a poor mathematician Apr 18 '12 at 2:31
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    $\begingroup$ Functions are rules that associate every valid input to one and only one valid output. Equations are expressions that express an equality (hence "equation", get it?) between two things. The lines, planes, circles, etc. consist of all points that "satisfy the equation" (that, when plugged into the equation, yield a true statement about numbers). A function would be a relation between inputs and outputs, which is something different. (2) Because every equation can be rewritten so that it is an equation "equal to $0$"; that provides uniformity, and it also makes certain properties clearer. (cont) $\endgroup$ – Arturo Magidin Apr 18 '12 at 4:02
  • $\begingroup$ (cont) E.g., it is easier to see what are the points that satisfy the equation $(x-1)(x-2) = 0$ than to see what are the points that satisfy the equation $x^2+2 = 3x$, even though both equations correspond to the same set of points. $\endgroup$ – Arturo Magidin Apr 18 '12 at 4:02
  • $\begingroup$ @ArturoMagidin Thanks!! I do know the difference between a function and an equation. Thanks for explaining "equal to zero" part. I wanted to know if it's possible to use only functions for defining curves, planes, lines, circles etc. But it seems like J.M. answered that question... $\endgroup$ – DrStrangeLove Apr 18 '12 at 4:13
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Compilation of Arturo Magidin's comments:

(1) Functions are rules that associate every valid input to one and only one valid output. Equations are expressions that express an equality (hence "equation", get it?) between two things. The lines, planes, circles, etc. consist of all points that "satisfy the equation" (that, when plugged into the equation, yield a true statement about numbers). A function would be a relation between inputs and outputs, which is something different.

(2) Because every equation can be rewritten so that it is an equation "equal to 0"; that provides uniformity, and it also makes certain properties clearer. E.g., it is easier to see what are the points that satisfy the equation $(x−1)(x−2)=0$ than to see what are the points that satisfy the equation $x^2+2=3x$, even though both equations correspond to the same set of points. –

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