Lecture-13-b - Lecture 13 (supplement) These notes exhibit...

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Lecture 13 (supplement) These notes exhibit a few ways of showing that: a) every line in the plane is the graph of a linear equation, and a) the graph of every linear equation is a line? The Frst demonstration, which is based on similar triangles, follows a pattern that can be found in expositions of analytic geometry in the early 18th century. I have found this argument repeated in algebra textbooks from the 1970s. The fat Algebra textbook of Sullivan, which was formerly used for Math 1021 at LSU, has a version of this, but it’s not presented carefully, and there is much hand-waving. Theorem 1. Suppose that P 0 = ( x 0 , y 0 ) and P 1 = ( x 1 , y 1 ) are points in the plane with x 0 n = x 1 . Let L be the line through P 0 and P 1 , and let m = y 1 y 0 x 1 x 0 . Then, for any point P = ( x, y ) in the plane: P is on L if and only if y y 0 = m ( x x 0 ) . (1) Proof. If y 1 = y 0 , then equation (1) reads y = y 0 and L is the line perpendicular to the y -axis through (0 , y 0 ). The conclusion in this case is immediate from the deFnition of the coordinate system. Let us turn to the situation when y 1 n = y
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This note was uploaded on 11/23/2011 for the course MATH 6302 taught by Professor Madden during the Summer '11 term at LSU.

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Lecture-13-b - Lecture 13 (supplement) These notes exhibit...

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