1. Slide 1 Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome BackgroundsCommunication!

2. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Communication Game Board Category 1 Category 2 Category 3 Category 4 Category 5 100 200 300 400 500 Final Challenge Final Challenge

3. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Solutions for 100 The point ( 1, -3 ) Is a solutions of the equation: 2x – 3y = 6. False. It is not a solution! 2(1) – 3(-3) = 6 2 –(-9) = 6 2 + 9 = 6 11 = 6

4. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Solutions for 200 The point ( 2, -5 ) is a solution of the system of equations: 7x + 5y = -6 y = -7/5x + 2 7x + 5(-7/5x +2) = -6 7x + -7x + 10 = -6 10 = -6 NO – inconsistent system These lines are parallel!!

5. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Solutions for 300 This system of equations: 7x + 5y = 10 y = -7/5x + 2 has this many solutions 7x + 5(-7/5x + 2) = 10 7x – 7x + 10 = 10 10 = 10 (true!) I nfinitely many - they are the same line!

6. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Solutions for 400 A store sells packs of AA batteries for $1.89 and packs of AAA batteries for $1.53, tax included. Jeff bought 8 packs and spent a total of $14.04. This is the number of packs of each kind Jeff can buy. x + y = 8 1.89x + 1.53y = 14.04 -1.89x-1.89y = -15.12 -.36y = -1.08 y = 3 x = 5 3 packs of AAA and 5 packs of AA?

7. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Solutions for 500 A tank was filled with water at the rate of 1.5 gal / min. By 10 a.m., ti contained 50 gal. Several minutes later the water going into the tank was turned off, the plug pulled and the tank drained at a rate of 2.3 gal / min. The tank was empty at 11:08 a.m. This is the number of minutes it took to drain the tank 50 + 1.5x = 2.3 ( 68 – x ) 50 + 1.5x = 156.4 – 2.3x 1.5x + 2.3x = 156.4 – 50 3.8x = 106.4 x = 28 68 – 28 = 40 40 minutes?

8. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Inequalities for 100 The y-axis, for y > 1, is a solution to the inequality 2x – y > 2 NO! The point (0,2) is on the y-axis for y > 1. By substitution, 2(0) – 2 > 2 -2 > 2 is false.

9. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Inequalities for 200 The solution to a system of inequalities is represented on a graph by this. The shaded region of the graph

10. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Inequalities for 300 This must be true in order for an ordered pair to be a solution to a system of linear inequalities. It must satisfy every inequality in the system.

11. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Inequalities for 400 You plan to buy a pair of jeans for $25 and some T-shirts for $12 each. You have only $60 to spend. These are the two inequalities that represent the number of T-shirts you can buy. 12x + 25 ≤ 60 x = # of T-shirts

12. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Inequalities for 500 AnswerQuestion?

13. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Linear Programming for 100 You are given an equation and several inequalities. The equation is called the _______________ and the inequalities are known as _______________ objective function constraints

14. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Linear Programming for 200 Find the minimum and maximum values of the objective function given the vertices of the feasible region are (0,40) ; (30,30) ; (40,0) ; (0,0) with C = x – y min of -40 at (0,40) and max of 40 at (40, 0)

15. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Linear Programming for 300 Find the minimum and maximum values of the objective function given the following constraints: x ≥ 0 y ≥ 0 x + y ≤ 9 C = 2x + 3y min of 0 at (0,0) and max of 27 at (0,9)

16. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Linear Programming for 400 These are the minimum and maximum values of C= 7x + 5y for the given constraints: x ≥ 1 y ≥ 2 x + 3y ≤ 13 and x + y ≤ 7 minimum of 17 at (1,2) maximum of 45 at (5,2) With test points of intersection at (1,2), (1,4), (4,3) and (5,2)

17. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Linear Programming for 500 A landscaper uses two brands of fertilizer, each containing different amounts of phosphates and nitrates. Brand X contains 6 oz of phosphates per package; Brand Y contains 5 oz. The lawn requires at least 30 oz of phosphates. Brand X contains 5 oz of nitrates, Brand Y, 2.5 oz per package. The lawn requires at least 20 oz of nitrates. One package of brand X costs $7.98 and one of Brand Y is $3.42. Write the constraints of the problem and the objective function. x ≥ 0 y ≥ 0 6x + 5y ≥ 30 5x + 2.5y ≥ 20 P = 7.98x + 3.42y x ≥ 0 y ≥ 0 6x + 5y ≥ 30 5x + 2.5y ≥ 20 P = 7.98x + 3.42y

18. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Eliminate/Substitute for 100 Why is substitution a good choice to solve the following system? y = 2x -3 3x – y = 4 One of the variables (y) is isolated already.

19. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Elimination/Substitution for 200 Why is elimination a good method for solving the following system of equations? 3x – 4y = 2 3x + 4y = 4 The y variables have coefficients that are additive inverses. (they add to zero!)

20. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Elimination/Substitution for 300 Solve the following system using substitution: 2x + 3y = 5 x + 5y = 20 2(-5y + 20) + 3y = 5 -10y + 40 + 3y = 5 -7y = -35 y = 5 x + 5(5) = 20 x = -5 (-5,5)

21. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Elimination/Substitution for 400 A small cheese pizza costs $5 while a small cheese, sausage and mushroom costs $8. Mrs. Peterman spent $82 on a pizza party for her son. The number of cheese pizzas was 2 greater than twice that of the cheese and sausage. How many of each did she buy? C = # of cheese pizzas S = # of sausage pizzas 5C + 8S = 82 C = 2 + 2S 5(2 + 2S) + 8S = 82 10 + 10S + 8S = 82 18S = 72 S = 4 C = 10 Check: 5(10) + 8(4) = 82 50 + 32 = 82 82 = 82 (yeah!!!!)

22. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Elimination/Substitution for 500 You are buying beads and string to make a necklace. The string costs $1.50. A package of 10 red beads costs $.50 and a package of 25 silver beads costs $.75. You only have $7.00 and you need 150 beads. How many of each type should you buy? x = silver beads y = red beads 25x + 10y = 150.75x +.50y + 1.50 = 7.00 3(25x + 10y = 150) 75x + 30y = 450 75x + 50y = 550 20y = 100 y = 5 x = 4 Check:.75(4) +.50(5) + 1.50 = 7.00 7.00 = 7.00 (it works!!)

23. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Grab Bag for 100 The multiplicative inverse of -4 is this -¼ some of you thought it was 4, didn’t you?

24. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Grab Bag for 200 Name the coefficients and constants of the expression: 3x² - 6x + 4 coefficients: 3, -6 constant: 4

25. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Grab Bag for 300 A solution to a system of linear equations has one ordered pair. What does this tell us about the graph of the equations? The two equations graph as lines that intersect at that point!

26. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Grab Bag for 400 The x-intercepts of a particular function are 3 and -2. What do these numbers represent? The coordinates where the graph of the function intersect the x-axis.

27. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Grab Bag for 500 Arrange the sets of numbers from smallest to largest… integers, whole numbers, natural numbers, rational numbers, real numbers natural numbers whole numbers integers rational numbers real numbers (believe it or not, there are infinite sets that are larger than others!!)

28. Copyright © 2004 Glenna R. Shaw & FTC Publishing Background Courtesy of Awesome BackgroundsAwesome Backgrounds Game Board Game Board +100 -100 0 +100 -100 0 +100 -100 0 Reset Scores Final Challenge! AnswerQuestion? Write Your Final Challenge Wager