MATH 106 QUIZ 3 Instructor: E. Cooper NAME: _______________________________ INSTRUCTIONS The quiz is worth 100 points. There are 8 problems. This quiz is open book and open notes. This means that you may refer to your textbook, notes, and online classroom materials, but you must work independently and may not consult anyone (and confirm this with your submission). Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also. In your document, be sure to include your name and the assertion of independence of work. 1. (7 pts) The matrix below is the final matrix for a system of two linear equations in the variables x1, and x2. What can be concluded about the solution of the system? (no explanation required) 1. _______ 1 3 7 [ | ] 0 0 0 A. B. C. D. There is no solution. The unique solution to the system is x1 = 3 and x2 = 7. There are infinitely many solutions. The solution is x1 = 3t + 7 and x2 = t, for any real number t. There are infinitely many solutions. The solution is x1 = 3t + 7 and x2 = t, for any real number t. 2. (8 pts) The shaded (purple) portion of the graph is the solution set of which inequality? 2. ___ Show work or explain how you have made your choice. A. y 4x 8 B. y 4x + 8 C. y x 8 D. y 4x + 8 3. (8 pts) Which of the following points is in the feasible region of the system 3. _______ of inequalities? That is, which of the following points satisfies all three inequalities below? y 3x 2y x+y Show work or explain how you have made your choice. A. B. C. D. (3, 4) (6, 1) (8, 5) (9, 2) 4. (8 pts) State the result of performing the row operation ( 3)R2 + R1 R1 on the matrix [ 7 3 0 | ] 2 1 3 Note that you are not asked to solve the system. Instead you are asked to perform a particular row operation. 5. (11 pts) Given the following application, define two variables, express your answer to the question as a linear inequality, and also state appropriate nonnegative restrictions. (You are not asked to produce a graph.) Molly wants to purchase ribbon to make bows. Red ribbon costs $1.59 per yard and gold ribbon costs $2.70 per yard. How many yards of each type of ribbon can Molly purchase if she can spend at most $20 altogether? Template for problems involving augmented matrix methods: [ 0 0 0 0 | ] 0 0 (If you wish, for problems #6 and #7 , you can use the Equation Editor template shown above, to make the typing easier. Just copy, paste where appropriate, and edit with the Equation Editor as needed. Or, you are welcome to use a table or a spreadsheet, or to hand-write and scan.) 6. (17 pts) Solve the system by using augmented matrix methods. Show your steps (row operations) in a similar way to the examples in the book in section 4.2. To avoid typing subscripts, I am labeling the variables x and y but if you prefer, you can label them x1 and x2. 2x + 6y = 14 x + 3y = 7. (17 pts) Solve the system by using augmented matrix methods. Show your steps (row operations) in a similar way to the examples in the book in section 4.2. 5x 10y = 4x + 8y = 8. (24 pts) Consider the following graph of a feasible region, which is bounded by the lines x + 3y = 5, x - 3y = 3, x = 0, and y = 0, as indicated. (a) Write the system of inequalities corresponding to the graph of the feasible region. (For each , replace by either or , as appropriate.) x + 3y x - 3y 5 3 x 0 y 0 (b) Find the coordinates of the corner points A, B, C, and D of the feasible region. For each of the four corner points, state which system of equations must be solved, and show how to solve it. (Continue on another page for more space.)