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Is it necessary to rearrange the equation of a line so that it is in the $y=mx+b$ form before using substitution to check whether a point is on the line? If yes, why? If no, why?

Lucille
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  • Why would rearranging it make a difference to which points satisfied it? – almagest Sep 08 '14 at 17:29
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    Do not rearrange! Every rearrangement is an opportunity to make a mistake. Take for example the nice $3x-4y+27=0$, If I rearrange to the form $y=mx+b$, those minus signs may cause trouble, and nice integers turn into ugly fractions. – André Nicolas Sep 08 '14 at 17:45
  • @Andre however for many other problems apart from this one, writing $y=mx+c$ is a very good idea. – JP McCarthy Sep 09 '14 at 14:43
  • Indeed there are no universal rules. But I have often seen standard not needed manipulations leading to unnecessary complication, and all too often, error. – André Nicolas Sep 09 '14 at 14:52

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No, you just need to ensure, upon substitution, that equality holds.

$$y = mx + b \iff y - mx = b \iff y - mx - b = 0 $$

If $(x_0, y_0)$ is a solution to any one of the above, it is a solution to all the equations above. It is also a solution to $y - y_0 = m(x - x_0)$

Adding/subtracting a term to/from each side of an equation doesn't change the equality, and multiplying/dividing the equation by a non-zero constant doesn't change the equality.

If you know $x= a$ but need to solve for $y$, then using the form $y = mx + b$ and substituting for $x$, gives you the value of $y$ most immediately. But given $x$, we can solve for $y$ no matter what form the equation of a line.

amWhy
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