Quiz1Solutions

# Quiz1Solutions - a free variable 2 Find the general...

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Newberger Math 247 Spring 02 Quiz Solutions 1.1 and 1.2 1. Consider the following system. x 1 + x 2 + 4 x 3 = - 2 - 2 x 1 - 2 x 2 + hx 3 = k 2 x 2 + 8 x 3 = 0 Find all h and k such that the system has (a) no solution, (b) a unique solution, and (c) an inﬁnite number of solutions. Reduce the augmented matrix for the system to echelon form. 1 1 4 - 2 - 2 - 2 h k 0 2 8 0 1 1 4 - 2 0 2 8 0 0 0 h + 8 k - 4 . (a) This system has no solution when h = - 8 and when k is any real num- ber except 4 (because this will put a pivot position in the last column of the augmented matrix). (b) It has a unique solution when k is any real number, and h is any real number except - 8 (because this will put a pivot position in every column of the coeﬃcient matrix). (c) It has an inﬁnite number of solutions when h = - 8 and k = 4 (because then it will have a column that does not have a pivot position, and hence
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Unformatted text preview: a free variable). 2. Find the general solution of the following system of equations. x 2 + 3 x 3 =-2-2 x 1-3 x 2-5 x 3 =-4 x 1 + 2 x 2 + 4 x 3 = 1 We write the augmented matrix for this system and reduce it to reduced echelon form. 1 3-2-2-3-5-4 1 2 4 1 ∼ 1 2 4 1 1 3-2-2-3-5-4 ∼ 1 2 4 1 0 1 3-2 0 1 3-2 ∼ 1 2 4 1 0 1 3-2 0 0 0 ∼ 1 0-2 5 0 1 3-2 0 0 . So the basic variables are x 1 and x 2 , and x 3 is free. The equations are: x 1-2 x 3 = 5 x 2 + 3 x 3 =-2 In general form the solution set is given by: x 1 = 2 x 3 + 5 x 2 =-3 x 3-2 x 3 is free....
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