Week 1 (8/24/2004-8/30/2004) Read 2.1: Solution of linear system by the Echelon method

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1 Week 1 (8/24/2004-8/30/2004) Read 2.1: Solution of linear system by the Echelon method Important Terms and Concepts: Possibilities for the solutions of a system of two linear equations in two unknowns (see page 50) The Echelon method (see the method, example 1, example 2 and example 3 on pages 51-53) Steps for solving a system of linear equations using Echelon method (see the steps on page 54) Applications (see examples 5 and 6 on pages 55-56) Assignments: 15, 25, SET UP ONLY problems 34, 36, 37, 38, 39, 42, and 43 on page 58. Here are some problems from the same 2.1 Exercises solved to help you do this assignment. This is problem #1 on page 57: x + y = 9 (1) 2x y = 0 (2) Multiply equation (1) by -2 and add the result to equation (2). The new system is x + y = 9 (1) -2R 1 + R 2 R 2-3y = -18 (3) Multiply equation (3) by to get x + y = 9 (1) R2 R 2 y = 6 (4) Substitute 6 for y in equation (1) to get x = 3. So the solution is (3, 6). This is problem 23 on page 57: 2x + y z = 5 (2) x y + z = -2 (3) Eliminate x in equations (2) and (3). -2R 1 + R 2 R 2 - y 3z = 1 (4)

2 -1R 1 + R 3 R 3-2y = - 4 (5) Eliminate y in equation (5) - y 3z = 1 (4) -2R 2 + R 3 R 3 6z = -6 (6) Make the coefficient of the first term in each equation equal 1. -1R 2 R 2 -y 3z = 1 (7) 1 R3 R 3 6 z = -1 (8) Substitute -1 for z in equation (7) to get y = 2. Finally, substitute -1 for z and 2 for y in equation (1) to get x = 1. The solution is (1, 2, -1). Here is problem #35 on page 58: Let x = the cost per pound of rice y = the cost per pound of potatoes 20x + 10y = (1) 30x + 12y = (2) Problem #40 on page 58 x = the number of fives y = the number of tens z = the number of twenties Since the number of fives is three times the number of tens x + y + z = 70 x -3y = 0 5x + 10y + 20z = 960 x = 3y

3 Problem #41 on page 58 x = the amount invested at 6.5% in mutual funds y = the amount invested in government bonds at 6% z = the amount invested in the bank at 5% Since she invested $10,000 among these 3 accounts, one equation will be x + y + z = 10,000 Since she invested twice as much in bonds as she did in mutual funds, another equation is y = 2x. Since her return on the investment is $605, the last equation is.065x +.06y +.05z = 605 x + y + z = 10,000 (1) 2x - y = 0 (2).065x +.06y +.05z = 605 (3) Week 2 (8/31/2004-9/6/2004) Read 2.2: Solution of Linear systems by the Gauss-Jordan Method Important Terms and Concepts: Definition of a matrix (see pages 59-60) Row operations (see the steps on page 60) Gauss-Jordan method: calculating by hand (see example 2 on page 62) Gauss-Jordan method: calculating by graphing calculator (see the same example above on page 64) Systems with nonunique solutions (see example 3 on page 66) Systems with an infinite number of solutions (see example 4 and example 5 on pages 66-67) Application (see example 6 on page 68) Steps for solving a system of linear equations using matrices (see the steps on page 70) Reduce row echelon form (rref) method to solve systems of linear equations on the TI-83(Your graphing calculator manual contains useful information on how to work with matrices on your calculator) Assignments: 18, 24, 26, 28, 32, 44, 48, 50, 60, and 62 pages To solve each application you need to do the following: a. Identify the variables clearly including correct units. b. Write the mathematical model (write the equations).

4 c. Solve the system( use your graphing calculator ( rref command)) d. Write the solution as a complete sentence using correct units. Here are some problems from the same 2.2 Exercises solved to help you do this assignment. This is problem #17on page 71 x + y = 5 x y = -1 The system has augmented matrix Use row operations as follows 1 5-1R 1 + R 2 R R2 R R 2 + R 1 R Read the solution from the last column of the matrix. The solution is (2, 3). This is problem #43on page 71 Let x = the number of hours to hire the Garcia firm y = the number of hours to hire the Wong firm 10x + 20y = x + 10y = 750 5x + 10y = 250 The augmented matrix of the system is

5 Using rref command from TI-83 graphing calculator, we have the following matrix Read the solution from the last column of the matrix. The solution is (20, 15). Hire the Garcia firm for 20hr and the Wong firm for 15hr. Answers To Even problems: 18. The solution is (-3, 4) 24. (There are an infinite number of solutions, each of the form (y +1, y), for any real number y.) 26. The solution is (-1, 2, -2) 28. (There are an infinite number of solutions, each of the form (z -3, -z + 9, z), for any real number z.) 32. The system is inconsistent and has no solution. 44. The solution is (22, 56, 22). There were 22 units ordered from Toronto, 56 units ordered from Montreal, and 22 units ordered from Ottawa. 48. The solution is (100, 50, 50). The company should buy 100 vans, 50 small trucks, and 50 large trucks. 50. The solution is (12, 8, 16, 0). Therefore, 12 cars should be sent from I to A, 8 cars from II to A, 16 cars from I to B, and no cars from II to B.

6 60. The solution, which may vary slightly, is 2340 of the first species, of the second species, and 224 of the third species. ( All of these are rounded to the nearest whole number.) 62. (a) There is no solution o the system. Therefore, it is not possible to utilize all resources completely. (b) The solution is (150, 50, 20). Therefore, allot 150 acres for honeydews, 50 acres for onions, and 20 acres for lettuce.