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by: Addison Beer


Addison Beer
GPA 3.76


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Class Notes
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This 5 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 98 at University of Washington taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/192087/math-98-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15
Line Through Two Points ln The Plane Math 98 Lines It is easy to write the equation of the unique line that connects two points in the plane but you have to do it right Herels two ways even though they are essentially the same you have two points in the plane say ab and cd it is easy for example from a picture to nd the slope of the line connecting them indeed that has to be d 7 b b 7 d m 7 7 Cid a7c Since a line has an equation of the form y mm k we only need to nd 16 and that we can do by forcing the line to go through one of the point the pother one will also follow since we have picked the right slopei For example we can plug in z a y b and see that k must be such that bmak ak a7c kbiab7dba7c7ab7d ab7bc7abadad7bc a7c a 7 c a 7 c a 7 c and the equation now becomes ymzk zad7lcb7dzad7bc 1 a 7 c a 7 c a 7 c You really would not want to memorize It is much faster to retrace the steps we made to nd kl Another way is to write a linear function that will become an identity when we plug in the two pairs describing the points One standard way to do this is to make sure we get 0 0 when say I ay bi This means that the function must look like Ay7b Bz7a for some numbers A and B Now we make sure that the two sides are both equal to say 1 when I cy di lief we need Ad7blBc7al that is 1 1 147 3 7127 57a so that the equation is yib 17a 2 deb 57a Of course our choice of 0 0 and 1 1 was just one of the many possible and we could also have started working with the other point rst This just underlines that there are several ways in which to write an equation for the same line Obviously 1 and 2 de ne the same line We can go from one to the other with a few algebraic steps For instance starting with 2 we can write deb yibciaria deb ybciaria y712ciad7bzia 57a yibcibaidabazdib 57a yi bcidazdib sea which is precisely 1 when you multiply both numerator and denominator by 71 Linear Programming Math 98 Linear inequalities The topic of linear programming is huge in applications7 and is not exactly the easiest part of our materiali Here is a short sketch of one possible approach 1 The Problem A linear programming problem looks like this we have linear function in a certain number of variables but we will stick to the simple case of two variables and a certain number of linear constraints on those variables We are looking for the point s that maximize or minimize the function7 within the set de ned by the constraints As an example7 suppose we have a set of constraints like oz 4 oyZO 02173y 5 01y2 and a target function given by 61 7 y that we want to maximize A good rst step is to draw a picture showing what the constrained set the area shaded in green is 5 2 Solving The Problem aymhkv mcmmw Miammmmw m qumuiwmmcmmm mqmmwmamwmmdmmmmm mum mmwmmMmmgmmmmmgmmmmm wawmmmmcmmws mmmmm 4mmmmmwmmmmwwm in m mmmymmmnmmwmmmm m A u


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