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First and last name: Open book (but don't use the method from the book), calculators allowed. 1. [10 points ] Solve the system of equations +4 = 13 4 ,2 + = 7 2 ,2 ,7 = ,19 by writing the corresponding augmented matrix and using elementary row operations. I want to see you write the steps that you perform, e.g., 2 , 3 1, to illustrate that you've understood the method. Please present your solution clearly, preferably in the same way that I use in lectures. x z x y x y z z R R The augmented matrix is (2 points) 0 1 0 4 13 1 @ 4 ,2 1 7 A 2 ,2 ,7 ,19 There is already a 1 in the top left position, so the row operations leading to zeros in the rst column are simply 0 1 0 4 13 1 1 0 4 13 1 7 to get @ 0 ,2 ,15 ,45 A 2 , 4 1 4 ,2 2 ,2 ,7 ,19 2 ,2 ,7 ,19 (1 point for correct operation, 1 point for correct arithmetic), then 0 1 0 4 13 1 1 0 4 13 0 ,2 ,15 ,45 to get @ 0 ,2 ,15 ,45 A ,7 ,19 0 ,2 ,15 ,45 3 , 2 1 2 ,2 (1 point for correct operation, 1 point for correct arithmetic). Finally, to get a zero in the third row/second column, do 0 1 0 4 13 1 1 0 4 13 0 ,2 ,15 ,45 to get @ 0 ,2 ,15 ,45 A ,7 ,19 0 0 0 0 3 , 2 2 ,2 (1 point for correct operation, none for arithmetic!). Now it's in echelon form, rewrite this as a system of equations (1 point): +4 = 13 ,2 ,15 = ,45 Use back substitution: solve for and in terms of to get = , 452 + 152 and = 13 , 4 (1 point). Hence the answer is 15 ) : 2 Rg ( ) = f(13 , 4 , 45 + 2 2 (2 points; you lose these two points if your solution does not convey the fact that may take any real value). Solution: R R R R R R x z y z y x x z y z x; y; z z; z; z z 1 z : z