Thread: ax+by=1 form transformations

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

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:

$\displaystyle ax + by = 1$

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

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

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

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