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When I study maths, I try to understand why the mathematicians brought out this concept or what usefulness they might have seen in the concept that they worked upon. So when I started with straight lines, what hit me was that why do we have to have so many forms of equations of straight lines? All they do is just help make more questions but don't really help us in real life applications of straight line concepts (or do they?). What was the need to explicitly define those forms? Why not only one general form was defined?

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  • $\begingroup$ Could you mention some examples? $\endgroup$ – zoli Jun 6 '15 at 11:27
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    $\begingroup$ Short answer: Each version of an equation of a line is useful to some people at some times. For example, I made a payroll spreadsheet calculating income tax for employees in Kenya and succeeded where many others failed because I used the point-point form of the equation of a line. $\endgroup$ – Rory Daulton Jun 6 '15 at 11:34
  • $\begingroup$ Which forms do you believe have no "real life applications"? Certainly all of the usual ones are useful in various contexts. $\endgroup$ – Travis Willse Jun 6 '15 at 11:36
  • $\begingroup$ For example, 3x+y-2=0 is the equation of a straight line in its general form then from this the fact that y= -3x+2( which is the slope intercept form) is implied and can be used as per our convenience. What made mathematicians define that explicitly as a totally different form? (looking from its mathematical applications point of view) $\endgroup$ – ModCon Jun 6 '15 at 11:36
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    $\begingroup$ The slope intercept form cannot be generalized to, say, three dimensions. Neither the $ax+bx+c=0$ could be generalized. ($ax+bx+cz+d=0$ would be a plane.) However, if $a=(u,v) $ and $b=(e,f)$, $z=(x,y)$ are vectors then $z=ta+b$ is a dimension independent form for straight lines. ($t$ is a real parameter). $\endgroup$ – zoli Jun 6 '15 at 11:48
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This started as a comment on an over-general and unspecific question, but got too long.

There are forms of equation which represent the fact that a line is defined by any two points on it, or by a single point and a direction. The point and direction form is useful in determining the tangent or normal to a differentiable curve.

Some forms distinguish between dependent and independent variables, others treat the variables as equivalent to each other. The vector form of a straight line generalises to multiple dimensions, and shows that the straightness of the line is preserved by a variety of transformations which preserve the essential vector structure. Some forms relate more naturally than others to the fact that the straight line is a geodesic in Euclidean space.

Sometimes it is useful to treat lines in projective co-ordinates, which are fully homogeneous. If there is a geometric structure in view, but no natural rectilinear co-ordinates, barycentric co-ordinates can come in handy.

Euclid did geometry involving straight lines without our notion of co-ordinates or equations - and the first form I mentioned - the line through two points, is a co-ordinate form of what Euclid meant by a line - and so relates co-ordinate geometry to Euclid's propositional geometry.

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