ME 349 - Fonseca Fall 2016 Homework 7 Due on November 1st 2016 1. Solve the following systems of linear equations using Cramer's Rule. Use Matlab to compute the associated matrices' determinants. Provide all necessary printouts and show all computations. a. 3 x1 +6 x2 + 4 x3= -14 -2x1 + 5 x2 + 5 x3 = -33 +4x2 + 6 x2 - 3 x3 = 24 b. -5 x1 +3 x2 - x3 - 4 x4 = 15 +5x1 + 5 x2 - 2 x3 + x4 = 22 +3 x1 -5 x2 + 2x3 +5 x4 = -8 - 3x1 + 2 x2 -5 x3 - 7 x4 = 32 2. Solve the following systems of linear equations by hand using Gauss elimination. Show all your work. Check your answers using GaussNaive.m a. 4x1 -5x2 + 7x3 = -225 -12x1 -3x2 -19x3 = 536 5x1 +20x2 + 17x3 = -207 b. 5x + 13y - 15z - 17w = 149 -7x -19 y +5z -9w = -35 4x -2y +17z -5w = -51 -11x +25 y +13z +17w = -76 3. Put each one of the problems below in matrix form. Solve them through appropriate matrix operations via Matlab. a. A total of $50,000 is invested in three funds paying 6%, 8%, and 10% simple interest. The yearly interest is $3,700. Twice as much money is invested at 6% as invested at 10%. How much was invested in each of the funds. b. In an industrial process, water flows through three tanks in succession as illustrated in the figure. The tanks have unit cross-section and have heads of water x, y and z respectively. The rate of inflow into the first tank is u, the flow rate in the tube connecting tanks 1 and 2 is 6(x-y), the flow rate in the tube connecting tanks 2 and 3 is 5(y-z), and the rate of outflow from tank 3 is 5z. Assuming the system at steady state, solve for the level of water (x, y, z) in each tank. c. Lawnco produces three grades of commercial fertilizers. A 100-lb bag of grade-A fertilizer contains 18 lb of nitrogen, 4 lb of phosphate, and 5 lb of potassium. A 100-lb bag of grade-B fertilizer contains 20 lb of nitrogen and 4 lb each of phosphate and potassium. A 100-lb bag of grade-C fertilizer contains 24 lb of nitrogen, 3 lb of phosphate, and 6 lb of potassium. How many 100-lb bags of each of the three grades of fertilizers should Lawnco produce if 26,400 lb of nitrogen, 4900 lb of phosphate, and 6200 lb of potassium are available and all the nutrients are used? d. Provide the system of linear equations needed to find the values of the different traffic flows in matrix form. Solve for x1, x2, x3 and x4