midterm_solutions_06

midterm_solutions_06 - Math 133 Midterm Solutions. Problem...

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Unformatted text preview: Math 133 Midterm Solutions. Problem 1: The problem is asking to find all the values of a and b so that a given system system of equations has either no, a unique, or infinitely many solutions. Perform Gaussian row-reduction: 2- 1 1 1 2- a 4 3- 1 6 11 b 1 2- a 2- 1 1 4 3- 1 11 6 b 1 2- a- 5 2 a + 1- 5 4 a- 1 11- 16 b- 44 1 2- a- 5 2 a + 1 2 a- 2 11- 16 b- 28 . In the first step we interchange the first two rows; in the second step we subtract twice the first row from the second row, and subtract four times the first row from the third row; in the third step we subtract the second row from the third row. In principle we can also divide the second row by- 5 to get an explicit 1 in the second column. If 2 a- 2 6 = 0 (in other words a 6 = 1) then we can divide the third row by 2 a- 2 and obtain a 1 in the third column. Hence the system has a unique solution regardless of what values are on the right. (Indeed, we can uniquely solve for x 3 using the third equation, then we can uniquely solve for x 2 using the second equations, and finally we can uniquely solve for x 1 using the first equations.) If 2 a- 2 = 0 AND b 6 = 28 then the system is inconsistent: the third equation is of the form 0...
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midterm_solutions_06 - Math 133 Midterm Solutions. Problem...

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