ax+by=1 form transformations

**theimcq** · Jun 10th 2008, 03:19 PM

Hi,

I'm doing some power studying, so yep, I'm going to be posting a lot of questions. I was just wondering, how do the values of a and b affect the equation ax+by=1? I've Googled some stuff and haven't found any basic list of rules. The only thing I have found (from a practice test and answer key) is that if a>0 and b>0, then the line will have a negative slope and not pass through the 3rd quadrant. Any other rules that I should know about?

Haha, the Pre-Algebra stuff went right out the window, as you can tell (up until this point in my life I've had Pre-Alg, Alg 1, Geometry, then just finished Alg II this past year). :P

Thanks!
imcq

**Reckoner** · Jun 10th 2008, 08:25 PM

Originally Posted by theimcq

I was just wondering, how do the values of a and b affect the equation ax+by=1?

You may find it easier to answer this if you rewrite it in a more common form:

$ax + by = 1$

$\Rightarrow by = 1 - ax\Rightarrow y = \frac1b - \frac abx$

The slope-intercept form of an equation of a line is $y = mx + b$ where the slope is $m$ and the y-intercept is $(0,\;b)$ . So, looking at our equation, we find that the slope of the line is $-\frac ab$ and the line will intersect the $y$ -axis at $\left(0,\;\frac1b\right)$ .

You can make a few observations from these facts. For example, as $\lvert b\rvert$ increases, the absolute value of the line's slope and y-intercept both decrease and the line becomes flatter and closer to the $x$ -axis.

**theimcq** · Jun 11th 2008, 05:27 AM

Thanks so much! I totally forgot about changing it to the mx+b form. Case closed!

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