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Problem Set #2 Note: Use RREF on your calculator, unless you need to do by hand, as in #1 and #3. 1. Solve the system of linear equations by row reducing the augmented matrix by hand to convert it to row-reduced echelon form. Show all steps and label each elementary row operation. 3x 7 y z 11 2x y 2z 9 x 2 y 2 z 4 2. Solve each of the following systems of equations. For each system, give all solutions, or explain why none exist. If there are infinitely many solutions, give solutions as a linear combination of column vector(s): 7 x y 3z 2w 4 (a) 2 x 5 y z 3w 1 8 x 13 y 9 z 5w 5 2 x 4 y 3z 4w 7 (b) x 2 y z 5w 11 4 x 8 y 7 z 2w 5 3. Under what conditions on k does the system x 3y 7z 5 2x 7 y 6z 3 have xk yz k 2 (a) a unique solution (b) infinitely many solutions (c) no solutions 4. A rotated hyperbola has equation ax 2 bxy cy 2 d . Find an equation for the hyperbola if it is known that it has an x intercept at x 2 and the tangent line equation at x 1 is y 30 16 x . 7 7 5. Consider the electric circuit shown below. Given that I 1 15 85 115 , I2 and I 3 19 38 38 (amps) determine the resistances R1, R2 and R3 . If there are multiple solutions, restrict the value of the parameter(s) so that the solution makes physical sense. R1 B R3 A I1 F I3 10 V I2 15 V R2 C D E