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Math 141 WIR2_Dr. Rosanna Pearlstein 1. 1.Q, 2.1, 2.2 For the following quadratic equations, do the following: a) Determine whether the graph opens up or down. b) Find the vertex of the parabola. c) Find the maximum or minimum values. d) Find the x-intercepts, if any. I. 𝒚 = 𝟐𝟓𝒙𝟐 − 𝟏𝟎𝒙 + 𝟏 II. 𝒚 = 𝟕(𝒙 + 𝟒)𝟐 − 𝟏𝟎 2. If the demand price of an item is given by 𝑝(𝑥) = 180 − 3𝑥 (where x is the number of items) find the revenue function. a) What number of items produced yields maximum revenue? b) If the cost is 𝑐(𝑥) = 171𝑥 − 12, find the equilibrium point. 3. For each linear system below, use the Gauss-Jordan elimination method to determine the solution(s) or lack thereof. A. B. C. 3 𝑥 − 4𝑦 = −6 4 4𝑥 − 2𝑦 = −1 2 3 𝑥+ 𝑦=2 5 10 { 4 20 { 3𝑥 + 𝑦 = 3 5 2 {2 𝑥 + 3 𝑦 = 22 −2𝑥 + 𝑦 = 1 4. (From our textbook, Finite Mathematics by Tan, 2.1 #28). Box Office Receipts. A theater has a seating capacity of 900 and charges $4 for children, $6 for students, and $8 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $5600. How many children attended the show? 5. Which of the matrices below is in row-reduced form? A. B. 1 2 [0 0 0 0 6. 0 6 1 |5] 0 1 C. 1 [0 0 0 1 1 0 −3 0 |4] −1 1 1 [0 0 2 0 0 3 −7 1| 0 ] 0 1 Pivot each of the two matrices below about the boxed element. A. B. 1 [0 0 2 −6 5 8 6 9 |5] −1 3 3 [3 2 6 1 4 9 −3 0 |4] −1 1 7. Solve the two linear systems below, using the Gauss-Jordan elimination method as performed by “rref” on your calculator. 𝟐𝒙 = 𝒚 𝒚 − 𝟑𝒙 + 𝟐𝒛 = 𝟖 {𝟖𝒙 + 𝟑𝒚 = 𝟗 + 𝟐𝒛 { 𝟑𝒚 = 𝟏 + 𝟐𝒙 𝟐𝒛 = 𝟑 + 𝟐𝒚 − 𝒙 𝟒𝒙 = 𝟗 − 𝟓𝒚 8. Solve the linear systems below. Express the solutions using parameters. I. II. 𝟖𝒙 − 𝟒𝒚 − 𝟐𝒛 = −𝟒 { 𝒙+𝒚+𝒛= 𝟏 𝟏𝟎𝒙 − 𝟐𝒚 = −𝟐 { −𝟑𝒙 + 𝟐𝒚 − 𝒛 + 𝟒𝒘 = 𝟏 𝒙 + 𝒚 − 𝟓𝒛 − 𝒘 = 𝟐 9. Determine the value of k for which systems A and B below have infinitely many solutions and no solution respectively. A. B. 4𝑥 − 2𝑦 = −1 { −5𝑥 + 𝑘𝑦 = 5 4 4𝑥 − 𝑘𝑦 = −1 { −2𝑥 + 3𝑦 = 1 10. Comment on the following statements: are they true or false? Why? (a) A linear system with the same number of equations as unknowns always has a solution. (b) A linear system with more equations than the number of unknowns never has a solution. (c) A (2×2) linear system of two lines in the plane will have (at least) one solution if the two lines are not parallel. (d) A (3×3) linear system of three planes in space will have a solution if the three planes are not parallel. (e) A linear system with more variables than equations always has a solution. (f) A linear system with more variables than equations either has no solution or it has infinitely many solutions.