Question 332845
<font face="Garamond" size="+2">


Given a pair of equations of the form:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A_1x\ +\ B_1y\ =\ C_1]


And


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A_2x\ +\ B_2y\ =\ C_2]


If


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \exists\ k\ \in\ \mathbb{R}]


such that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A_1\ =\ kA_2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ B_1\ =\ kB_2]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ C_1\ \neq\ kC_2]


then the graphs of the two functions will be a pair of distinct parallel lines.


On the other hand, if


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \exists\ k\ \in\ \mathbb{R}]


such that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A_1\ =\ kA_2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ B_1\ =\ kB_2]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ C_1\ =\ kC_2]


Then the graphs of the two functions will be two instances of the identically same line.


Another way to look at it, and probably easier to tell just by looking:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{A_1}{A_2}\ =\ \frac{B_1}{B_2}\ \neq\ \frac{C_1}{C_2}\ \Leftrightarrow\ L_1\ \parallel\ L_2,\ L_1\ \neq\ L_2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{A_1}{A_2}\ =\ \frac{B_1}{B_2}\ =\ \frac{C_1}{C_2}\ \Leftrightarrow\ \ L_1\ \equiv\ L_2]





John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
</font>