Math 11 - Systems of Equations and Inequalities

Transcription

1 Name: Period: /30 ID: A Math 11 - Systems of Equations and Inequalities Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (1 point) What system of equations is represented by the following graph? A y = 0.7x + 2 y = 2.8x 2 4.9x + 2 B y = 0.7x + 2 y = 2.8x x + 2 C y = 2.8x 2 y = 0.7x x + 2 D y = 0.7x + 2 y = 2.8x x + 2 1

2 2. (1 point) Which graph represents the system of equations shown below? y = 0.9x 2 4x + 4 y = 2.1x A C B D 2

3 3. (1 point) Which graph represents the system of equations shown below? y = 0.2x 2 x + 3 y = 1.2x x 2 A C B D 3

4 4. (1 point) The graph of 9x + 7y < 5 is A C B D 4

5 5. (1 point) Which inequality represents the graph shown below? A y > 1 3 x 2 C y < 3x 1 2 B y > 3x 1 2 D y < 1 3 x 2 5

6 A banquet room can seat up to 450 guests. Each rectangular table seats 7 guests and each circular table seats 4 guests. 6. (1 point) If r represents the number of rectangular tables and c represents the number of circular tables, which graph represents the possible combinations of rectangular and circular tables? A C B D 6

7 7. (1 point) Which graph represents the solution to the inequality x 2 4x 6 > 8 A C B D 8. (1 point) Which graph represents the solution to the inequality x 2 + x A C B D 9. (1 point) A rectangle is x centimetres wide and 2x centimetres long. If the area of the rectangle has to be between 68 cm 2 and 90 cm 2, what are the possible values of x? A 2 34 x 6 5 C 34 x 3 5 B 34 x 45 D 68 x 90 7

8 10. (1 point) Which graph represents the solution to the inequality y x 2 + 3x 1 A C B D Short Answer 11. (1 point) Determine the number of points of intersection for the line y = 6x 17 and the curve y = 1.5x 2 3x 3.5. Supporting work must be shown to receive a mark. 8

9 12. (1 point) For what value of b will the line y = 6x + b and the curve y = 1.5x 2 3x intersect at only one point? 13. (3 points) Solve the following system of equations to the nearest tenth. y = 2x 2 5x + 3 y = 4x 2 + 2x (1 point) Solve this linear-quadratic system algebraically. y = 2x x y = 4 9

10 15. (2 points) Solve this quadratic-quadratic system algebraically. y = 2x 2 + 2x + 2 y = 5x 2 x (2 points) Solve this quadratic-quadratic system algebraically. y = (x 5) 2 y = 2x 2 + 8x (2 points) A path across a diagonal of a rectangular lot is 160 m long. If the perimeter of the lot is 448 meters, what are the dimensions of the lot? 10

11 18. (1 point) Graph the inequality: y 2x (1 point) For B( 3, b) to be a solution of 6x + 2y > 12, what must be true about b? Show your work. 20. (1 point) Create a quadratic inequality that has this solution: x 7 or x people will attend a concert if tickets cost $15 each. Attendance will decrease by 15 people for each $1 increase in the price. The concert promoters need to make a minimum of $ (1 point) What quadratic inequality represents this situation? 11

12 22. (1 point) Write an inequality to describe this graph. Problem 23. (3 points) The girls softball team is sponsoring a fundraising trip to see a professional baseball game. They charter a 62-passenger bus for $525. In order to make a profit, they will charge $17 per person if all seats on the bus are sold, but for each empty seat, they will increase the price by $1.50. What is the minimum number of passengers needed in order for the softball team not to lose money and how much would they have to be charged? 12

13 Math 11 - Systems of Equations and Inequalities Answer Section MULTIPLE CHOICE 1. B 2. D 3. D 4. D 5. C 6. B 7. A 8. D 9. C 10. B SHORT ANSWER 11. Equate the expressions and simplify. 1.5x 2 3x 3.5 = 6x x 2 9x = 0 Use the discriminant b 2 4ac to determine the number of solutions to this equation. Substitute a = 1.5, b = 9, and c = 13.5 b 2 4ac = ( 9) 2 4(1.5)(13.5) = 0 Since b 2 4ac = 0, there is one solution to the system 12. Equate the expressions and simplify. 1.5x 2 3x = 6x + b 1.5x 2 9x b ( ) = 0 There is a single solution (point of tangency) when the discriminant equals 0: ( 9) 2 4( 1.5)(0.5 b) = 0 b = The solutions are: (1.2, 0.1) and (0, 3) 14. The solution is: ( 1, 0) 15. The solutions are: (0, 2) and ( 1, 2) 16. The solutions are: (5, 0) and (1, 16) 1

14 17. Draw a diagram to visualize the situation. 2x + 2y = 448 x + y = 224 x 2 + y 2 = x 2 + y 2 = Substitute y = 224 x into the fourth equation and solve for x: x 2 + ( 224 x) 2 = x x + x 2 = x 2 448x = 0 x 2 224x = 0 ( x 128) ( x 96) Substitute these values into y = 224 x to solve for y. y = and y = = 96 = 128 The lot measures 128 m by 96 m. 2

15 Substitute the coordinates of B in the inequality. 6x + 2y > 12 6( 3) + 2(b) > 12 Solve for b b > 12 2b > 30 b > Sample answer: (x + 7)(x + 4) 0, or x x ( 15 + x) ( x) y 3x 4 PROBLEM 23. Income=Seats Sold Price paid per seat Seats Sold=62 n Price Paid Per Seat = n Profit = Income - Bus Cost To not lose money, profit must be greater than zero so ( 62 n) ( n) n n Solve for n using the quadratic formula, remembering that n 0: n = ( 1.5) ( 529) 2( 1.5) 56.9 Substitute n = 56 into n to get ( 56) = Therefore the minimum number of passengers is 6 at $ per person. 3