Math Midterm Problem

17rkearns

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OK SO I have to write an equation in standard form of a line that passes through 0,-7 and -7,0. I THINK the slope is -1.. but I don't know how to translate that to the ax+by=c format!

Also I have to solve these two equations for their intercepts
2x+y=13 and 5x-2y=19... I'm supposed to solve it by graphing but my equations look like this: Which is kind of hard to see but...ScreenSh00ter_20130205171952.jpg When I was given the answer, which was 3,5 as the intercept
 
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OK SO I have to write an equation in standard form of a line that passes through 0,-7 and -7,0. I THINK the slope is -1.. but I don't know how to translate that to the ax+by=c format!
Suppose you have the equation in slope intercept form with rational coefficients.

\(\displaystyle y = \dfrac{p}{q} + \left(\dfrac{r}{s}\right)x,\ where\ p,\ q,\ r,\ s\ are\ integers\ and\ q,\ s \ne 0 \implies\)

\(\displaystyle qsy = ps + qrx \implies qrx - qsy = - ps.\)

\(\displaystyle Let\ u = - 1\ if\ qr < 0\ and\ u = 1\ if\ qr \ge 0 \implies qrux - qsuy = - psu.\)

\(\displaystyle Let\ t = the\ largest\ positive\ integer\ that\ factors\ qr,\ qs,\ and\ ps \implies \left(\dfrac{qru}{t}\right)x + \left(\dfrac{-qsu}{t}\right)y = \dfrac{-psu}{t} \implies\)

\(\displaystyle Ax + By = C,\ where\ A = \dfrac{qru}{t}, B = \dfrac{- qsu}{t},\ and\ C = \dfrac{- psu}{t}.\)

A will be a non-negative integer. B and C will be integers. And A, B, and C will have no common factor greater than 1.

Please do not memorize that as a formula. It is just the result of some basic algebraic manipulations.
 
I think that the simplest way to do this is to recognise that 'standard form' is "ax+ by= c" and put in the given values for x and y. If the line goes through (0, -7) then x= 0, y= -7 satisfies the equation: a(0)+ b(-7)= -7b= c. If the line goes through (-7, 0) then x= -7, y= 0 satisfies the equation" a(-7)+ b(0)= -7a= c. One of the first thing that tells us is that c= -7a= -7b so that a= b.

Of course, multiplying each or a, b, and c by any number will also give an equation for the line. Typically, what we want to do is choose the numbers so that a, b, and c are the simplest possible integers. Here, in order that a and b be integers, c must be divisible by 7. Taking c= 7, a= -1 and b= -1. That gives -x- y= 7. If you don't like those "-" signs, take c= -7 so that a= 1 and b= 1. That gives x+ y= -7 which we could also have got from -x- y= 7 by multiplying both sides by -1.
 
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