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July 13th, 2011, 03:03 PM   #1
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Converting a linear graph to a linear equation?

Ok so I'm really furious, I lost a good chunk of my mark on a test I just wrote, all because one of the problems asked for something I had never done before. It wanted me to convert a linear graph into a linear equation in standard form.

I have the slope-intercept form down pat, can write a graph to go along with it or vice versa no problem. Standard form is different though, I can write a graph out of a standard form equation, but I can't do it the other way around.

An example of slope-intercept form being: "y = 2/3 times X + 4"

While an example of Standard Form is: 3x + 2y = 6

Writing it as slope-intercept is logical because you just multiply X by the slope and then add the y-intercept. Standard form is different, no matter how hard I examine a graph, I can't figure out a way to write a linear equation in the form of AX + BY = C.
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July 13th, 2011, 03:32 PM   #2
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Re: Converting a linear graph to a linear equation?

Suppose you are given 2 points on a line: and and you are instructed to find the equation of the line written in standard form. One way to do this is to use the point-slope equation:



Compute the slope m: and we have:



Multiply through by



Now, rearrange into standard form:







Thus we see:







You can of course change the signs of all 3 values if needed to avoid most of them being negative.
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July 13th, 2011, 04:19 PM   #3
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Re: Converting a linear graph to a linear equation?

Thanks, that clears everything up nicely. I'm still furious that the test had that question though, I really hope the teacher has the good sense to not dock me any marks, 10% off my total mark is pretty steep for something they failed to cover or even mention before.
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July 13th, 2011, 04:56 PM   #4
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Re: Converting a linear graph to a linear equation?

Let's look at the definition of slope:

Given two points and



Now, suppose we let the second point instead be an arbitrary point on the line (x,y), we then have:



Rearranging, we find the point-slope formula for a line:



Now, suppose the first point is this gives us:



Solving for y, we obtain, the slope-intercept formula:



Now, suppose we know the x-intercept (a,0) and the y-intercept (0,b). We compute the slope:



Using the slope-intercept formula, we find the line is given by:





Divide through by b:



This is the two-intercept formula for a line. If we multiply through by ab, we have:



This is the line in standard form in terms of its intercepts.
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