HW1_solutions 167 - Math 167 homework 1 solutions October 6...

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Math 167 homework 1 solutions October 6, 2010 1.3.14a: Construct a 3 by 3 system that needs two row exchanges to reach a triangular form and a solution. There are many examples, but this is one: 3 y 2 z = 12 z = 3 4 x +3 y = 3 1.3.14b: Construct a 3 by 3 system that needs a row exchange to keep going, but breaks down later. 3 y 2 z = 12 4 x +3 y = 3 4 x +3 y = 1 1.3.18a: It is impossible for a system of linear equations to have exactly two solutions. If ( x,y,z ) and ( X,Y,Z ) are two solutions, what is another? Again, there are many solutions, i.e. any point on the line generated by the two given points. For example ( x + X 2 , y + Y 2 , z + Z 2 ) 1.3.18b: If 25 planes meet at two points, where else do they meet? These plane most also intersect at the line containing these two points. 1.3.20: Find the pivots and the solution for these four equations: 2 x + y = 0 x +2 y + z = 0 y +2 z + t = 0 z +2 t = 5 Using Gaussian Elimination we get 1
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2 x + y = 0 3 y +2 z = 0 4 z +3 t = 0 5 t = 20 So the pivots are 2 , 3 , 4 , and 5. Using back-substitution we find that ( 1 , 2 , 3 , 4) is the solution to the system. 1.3.26: Solve by elimination the system of two equations 2 x y = 0 3 x +6 y = 18 Draw a graph representing each equation as a straight line in the x - y plane; the lines intersect at the solution. Also, add one more line—the graph of the new second equation which arises after elimination.
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