Math 1070 Exam 2. March 16, 2006

Transcription

1 Math 1070 Exam 2 Name: March 16, Solve each of the following systems of equations using the indicated method: (a) Solve by graphing (include a sketch of your graph, indicating the graphing window, as well as the equations you graphed): 2x 1 3x 2 9 x 1 + 2x 2 8 I don t know how to get a graph in here, but two different possibilities for sets of equations work, which I show side by side. Solving for x 1 in the left column and x 2 in the right column would give 3 x 1 2 x x 2 3 x 1 3 x 1 2x x x The intersection points would be (1, 6) and (6, 1) respectively and, associating each coordinate to the correct variable, we get, in both cases, (b) Solve by substitution x 1 6 x 2 1 2x 1 + 5x 2 7 x 1 2x 2 4 From the second equation, x x 2, so the first equation is 2 (4 + 2x 2 ) + 5x 2 7, which simplifies to x Then, plugging this into x x 2, we get x ( 1 9 So x and x (c) Solve using elimination by addition 2x 1 + 3x x 1 6x 2 9 ) Add the second to the first to get 0x 1 3x 2 6. Then x 2 2. Substitute this into the first to get 2x , and thus x So x , and x

2 2. Do the following for each of the following matrices: State whether the matrix is in reduced form. If it is not perform an appropriate row operation to put it in reduced form. Then indicate whether the related system of equations has a unique solution, infinitely many solutions, or no solutions. (a) This satisfies all of the conditions of the reduced form. Further, since each column (except the constant terms) contains a leading one, we can conclude that there is a unique solution. (b) Not in reduced form because the leading 1 in the third row has 1 above it. Perform R 2 R 3 R 2 to get the reduced form: Since the last line corresponds to 0 1, there are no solutions. (c) Not in reduced form because the leading ones are not left to right as we go down. R 1 R 2 to get the reduced form: Perform Since we have a column with no leading ones and no line corresponds to 0 1, there are infinitely many solutions. 2

3 3. Consider the following matrices and perform the indicated operations (if possible) by hand: A B C D (a) BCA BCA B (CA) ( ) (b) 2D + A D + A (c) C 1 C

4 4. Consider the following system of equations: 3x 1 6x x 1 x 2 13 (a) Set up the augmented matrix for the system (b) Use Gauss-Jordan elimination to find the reduced row echelon form of this augmented matrix by hand. Show all work /3R R R R 1 R /3R R R R 2 R (c) Use the reduced row echelon form of the augmented matrix to determine the solution to the system of equations. x 1 5 x 2 3 4

5 5. Gizmos Incorporated manufactures three different gizmos: the Thingamajig, the Whatchacallit, and the Thing-bob. An overly simplified model of their production splits their manufacturing process into two steps: assembly and finishing. The following chart gives the time each step takes for each product. If the assembly department has a total of 300 hours available in a week and the finishing department has a total of 150 hours available in a week, what production schedule will allow Gizmos Incorporated to operate at full capacity? Production Step Thingamajig Whatchacallit Thing-bob Assembly Finishing The matrix for this comes right out of the table when we add the total hours available to the end of each row. We then find the reduced form (using a calculator) as follows: The reduced matrix corresponds to the following equations, where x 1, x 2, and x 3 represent the number of Thingamajigs, Whatchacallits and Thing-bobs produced respectively. Solving for x 1 and x 2, we get x 1 4x 3 0 x 2 + 3x x 1 4x 3 x x 3 Using the parameter t for x 3, we notice that t must be positive and that 3t must be less than 300, so that we have the following solution set: with 0 t 100, t an integer. x 1 4t x t x 3 t 5

6 6. Gizmos Incorporated, after a rapid expansion, operates plants in Cityburg and Townville. For various reasons, the hourly wages paid to employees differ at the two plants and for the different production steps, as given by the following table: Plant Assembly Finishing Cityburg $10.50 $11.00 Townville $13.00 $14.50 If we define matrix A as the matrix obtained directly from the table in question 5 and matrix B as the matrix obtained directly from the table in this question, (a) Compute the matrix BA (You may use a calculator). B and A , so that BA (b) Discuss the meanings of each entry in the matrix BA in terms of manufacturing. Each column represents a product (as it did in matrix A), and each row represents a plant (as it did in matrix B). The terms are determined by adding up the product of the hourly cost and hours needed in assembly and the product of the hourly cost and hours needed in finishing. Thus, the terms represent the total labor costs of manufacturing a given product (its column) at a given plant (its row). 6

7 7. The continuing saga of Gizmos Incorporated: Gizmos Incorporated discontinues production of the Thing-bob and reconfigures both of their plants to give the following hourly production schedule (entries indicating the number of units produced per hour): Plant Cityburg Townville Thingamajigs 10 8 Whatchacallits 5 8 The hours of operation for each plant are determined by the orders received each week. With the following orders, how long should each plant be operated to have the orders exactly filled? Thingamajigs Whatchacallits We are looking for six quantities: the hours for each plant and for each order. This corresponds to a 2 3 matrix. We have the following overall system, where c represents the hours of operation for Cityburg and t for Townville: c1 c 2 c t 1 t 2 t Then, multiplying both sides on the left by the inverse of the 2 2 matrix, we find that c1 c 2 c t 1 t 2 t Thus we have the following table for hours of operation for each order: Cityburg Townville