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I know,

$Ax + By = C$

is the equation of straight line but a different resource says that:

$y = mx + b$

is also an equation of straight line? Are they both same?

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    $Ax+By=C$ is sometimes called *general form*, *general linear form*, or *standard form*; $y=mx+b$ is often called *slope-intercept form*.2012-01-29

2 Answers 2

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Yes. That is, they both give the equation of a straight line and the equation of any non-vertical line can be written in either form.

If $B\ne 0$. Then you can write $Ax+By=C$ as $ By=-Ax+C $ and, since $B\ne0$, the above can be written $ y=-\textstyle{A\over B}x +{C\over B}. $

If $B=0$, the equation is $Ax=C$, which is a vertical line when $A\ne0$. In this case you can't write it in the form $y=mx+b$ (which defines a function).

On the other hand, given $y=mx+b$, you can rewrite it as $-mx+y=b$.




Note that for the equation $Ax+By=C$ with $A$ and $B$ both non-zero:

The $y$-intercept of its graph is $C/B$ and is found by taking $x=0$.

The $x$-intercept is of its graph is $C/A$ and is found by taking $y=0$.

The slope of the line is then $ {C/B-0\over 0-C/A } = -A/B$.

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$Ax + By = C$

$By = -Ax + C$

$y = -(A/B)x + C/B$

Let $m = -\frac{A}{B}$. Let $b = \frac{C}{B}$.

$y = mx + b$

So they are equivalent.

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    This was a good idea. It helps many people to illustrate equivalency in the most blunt way possible: Proving it.2012-01-26