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midterm-1-solns

# midterm-1-solns - Math 33A Spring 2008 First mid-term exam...

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Math 33A Spring 2008 First mid-term exam Name: Student ID: Instructions. There are five problems, each of which is worth 10 points. Answer each question. For full credit, show all work. Books, electronic devices, and notes may not be used. A piece of scratch paper is attached to the end of the exam. Problem Score 1 2 3 4 5 Total

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First mid-term exam 2 1. (10 points) Consider the following system of linear equations: y + 2 z = 1 2 x - y - 2 z = 7 x + y + 2 z = 5 . Perform Gauss-Jordan elimination on this system. If the system has any solutions, find all of them, and describe the solution set geometrically. If the system has no solution, state this. Solution: We form the augmented matrix A = 0 1 2 1 2 - 1 - 2 7 1 1 2 5 , and then perform the following row operations: 0 1 2 1 2 - 1 - 2 7 1 1 2 5 I III = 1 1 2 5 2 - 1 - 2 7 0 1 2 1 II - 2 I = 1 1 2 5 0 - 3 - 6 - 3 0 1 2 1 II ÷ - 3 = 1 1 2 5 0 1 2 1 0 1 2 1 I - II, III - II = 1 0 0 4 0 1 2 1 0 0 0 0 So x = 4, y + 2 z = 1. For a real parameter t , we set the free variable z = t , and see that there are infinitely
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