Notice that if you multiply the second equation by 2, the coefficients of the x terms are additive inverses.

Transcription

1 Use elimination to solve each system of equations. 2x y = 4 7x + 3y = 27 Notice that if you multiply the first equation by 3, the coefficients of the y terms are additive inverses. Now, substitute 3 for x in either equation to find y. The solution is (3, 2). 2x + 7y = 1 x + 5y = 2 Notice that if you multiply the second equation by 2, the coefficients of the x terms are additive inverses. Now, substitute 1 for y in either equation to find x. The solution is ( 3, 1). 4x +Manual 2y = - 14 esolutions Powered by Cognero 5x + 3y = 17 Page 1

2 The solution is ( 3, 1). 4x + 2y = 14 5x + 3y = 17 Notice that if you multiply the first equation by 3 and the second equation by 2, the coefficients of the y terms are additive inverses. Now, substitute 4 for x in either equation to find y. The solution is ( 4, 1). 9a 2b = 8 7a + 3b = 12 Notice that if you multiply the first equation by 3 and the second equation by 2, the coefficients of the y terms are additive inverses. Now, substitute 0 for a in either equation to find b. The solution is (0, 4). Page 2 A kayaking group with a guide travels 16 miles downstream, stops for a meal, and then travels 16 miles upstream. The speed of the current remains constant throughout the trip. Find the speed of the kayak in still

3 The solution is ( 4, 1). 9a 2b = 8 7a + 3b = 12 Notice that if you multiply the first equation by 3 and the second equation by 2, the coefficients of the y terms are additive inverses. Now, substitute 0 for a in either equation to find b. The solution is (0, 4). A kayaking group with a guide travels 16 miles downstream, stops for a meal, and then travels 16 miles upstream. The speed of the current remains constant throughout the trip. Find the speed of the kayak in still water. Let x represent the speed of the kayak and y represent the speed of river. The rate to travel down river would be with the current so the speed of kayak and speed of river are added. The speed of speed of time d kayak river (hours) current current d = rt x y = 2(x + y) x y = 4(x + y) Notice that the coefficients of the y terms are additive inverses, so add the equations. Page 3

4 The solution is (0, 4). A kayaking group with a guide travels 16 miles downstream, stops for a meal, and then travels 16 miles upstream. The speed of the current remains constant throughout the trip. Find the speed of the kayak in still water. Let x represent the speed of the kayak and y represent the speed of river. The rate to travel down river would be with the current so the speed of kayak and speed of river are added. The speed of speed of time d kayak river (hours) current current d = rt x y = 2(x + y) x y = 4(x + y) Notice that the coefficients of the y terms are additive inverses, so add the equations. So, the speed of the kayak is 6 mph. Steve subscribed to 10 podcasts for a total of 340 minutes. He used his two favorite tags, Hobbies and Recreation and Soliloquies. Each of the Hobbies and Recreation episodes lasted about 32 minutes. Each Soliloquies episodes lasted 42 minutes. To how many of each tag did Steve subscribe? Let x represent the number of Hobbies and Recreation podcasts and y represent the number of Soliloquies podcasts. Since Steve subscribed to 10 podcasts, then the number of Hobbies and Recreation and number of Soliloquies would equal 10. Each Hobbies and Recreation podcast talks 32 minutes. SO the total minutes for Hobbies and Recreation podcast is 32x. Each Soliloquies podcast talks 42 minutes. So the total minutes for Soliloquies podcast is 42x. Add Notice that if you multiply the first equation by 32, the coefficients of the x terms are additive inverses. Page 4

5 So, the speed of the kayak is 6 mph. Steve subscribed to 10 podcasts for a total of 340 minutes. He used his two favorite tags, Hobbies and Recreation and Soliloquies. Each of the Hobbies and Recreation episodes lasted about 32 minutes. Each Soliloquies episodes lasted 42 minutes. To how many of each tag did Steve subscribe? Let x represent the number of Hobbies and Recreation podcasts and y represent the number of Soliloquies podcasts. Since Steve subscribed to 10 podcasts, then the number of Hobbies and Recreation and number of Soliloquies would equal 10. Each Hobbies and Recreation podcast talks 32 minutes. SO the total minutes for Hobbies and Recreation podcast is 32x. Each Soliloquies podcast talks 42 minutes. So the total minutes for Soliloquies podcast is 42x. Add Notice that if you multiply the first equation by 32, the coefficients of the x terms are additive inverses. Now, substitute 2 for y in either equation to find x. Steve subscribed to 2 Soliloquies and 8 Hobby and Recreation podcasts. Use elimination to solve each system of equations. x +y = 2 3x + 4y = 15 Notice that if you multiply the first equation by 3, the coefficients of the x terms are additive inverses. Now, substitute 3 for y in either equation to find x. The solution is ( 1, 3). Page 5

6 Steve subscribed to 2 Soliloquies and 8 Hobby and Recreation podcasts. Use elimination to solve each system of equations. x +y = 2 3x + 4y = 15 Notice that if you multiply the first equation by 3, the coefficients of the x terms are additive inverses. Now, substitute 3 for y in either equation to find x. The solution is ( 1, 3). x y= 8 7x + 5y = 16 Notice that if you multiply the first equation by 5, the coefficients of the y terms are additive inverses. Now, substitute 2 for x in either equation to find y. The solution is ( 2, 6). x + 5y = 17 4x + 3y = 24 esolutions Manual by Cognero Notice that- ifpowered you multiply the first equation by 4, the coefficients of the x terms are additive inverses. Page 6

7 The solution is ( 2, 6). x + 5y = 17 4x + 3y = 24 Notice that if you multiply the first equation by 4, the coefficients of the x terms are additive inverses. Now, substitute 4 for y in either equation to find x. The solution is ( 3, 4). 6x + y = 39 3x + 2y = 15 Notice that if you multiply the first equation by 2, the coefficients of the y terms are additive inverses. Now, substitute 7 for x in either equation to find y. The solution is ( 7, 3). 2x + 5y = 11 4x + 3y = 1 Notice that if you multiply the first equation by 2, the coefficients of the x terms are additive inverses. Page 7

8 The solution is ( 7, 3). 2x + 5y = 11 4x + 3y = 1 Notice that if you multiply the first equation by 2, the coefficients of the x terms are additive inverses. Now, substitute 3 for y in either equation to find x. The solution is ( 2, 3). 3x 3y = 6 5x + 6y = 12 Notice that if you multiply the first equation by 2, the coefficients of the y terms are additive inverses. Now, substitute 0 for x in either equation to find y. The solution is (0, 2). 3x + 4y = 29 6x + 5y = 43 Notice that if you multiply the first equation by 2, the coefficients of the x terms are additive inverses. Page 8

9 The solution is (0, 2). 3x + 4y = 29 6x + 5y = 43 Notice that if you multiply the first equation by 2, the coefficients of the x terms are additive inverses. Now, substitute 5 for y in either equation to find x. The solution is (3, 5). 8x + 3y = 4 7x + 5y = 34 Notice that if you multiply the first equation by 7 and the second equation by 8, the coefficients of the x terms are additive inverses. Now, substitute 4 for y in either equation to find x. The solution is (2, 4). 8x + 3y = 7 7x + 2y = 3 Page 9

10 The solution is (2, 4). 8x + 3y = 7 7x + 2y = 3 Notice that if you multiply the first equation by 2 and the second equation by 3, the coefficients of the y terms are additive inverses. Now, substitute 1 for x in either equation to find y. The solution is (1, 5). 4x + 7y = 80 3x + 5y = 58 Notice that if you multiply the first equation by 3 and the second equation by 4, the coefficients of the x terms are additive inverses. Now, substitute 8 for y in either equation to find x. The solution is ( 6, 8). Page 10

11 The solution is (1, 5). 4x + 7y = 80 3x + 5y = 58 Notice that if you multiply the first equation by 3 and the second equation by 4, the coefficients of the x terms are additive inverses. Now, substitute 8 for y in either equation to find x. The solution is ( 6, 8). 12x 3y = 3 6x + y = 1 Notice that if you multiply the second equation by 2, the coefficients of the x terms are additive inverses. Now, substitute 1 for y in either equation to find x. The solution is (0, 1). 4x + 2y = 0 10x + 3y = 8 Page 11

12 The solution is (0, 1). 4x + 2y = 0 10x + 3y = 8 Notice that if you multiply the first equation by 3 and multiply the second equation by 2, the coefficients of the y terms are additive inverses. Now, substitute The solution is x in either equation to find y.. Seven times a number plus three times another number equals negative one. The sum of the two numbers is negative three. What are the numbers? Let x represent one number and y represent the second number. Notice that if you multiply the second equation by 3, the coefficients of the y terms are additive inverses. Now, substitute 2 for x in either equation to find y. Page 12

13 6-4 Elimination Using Multiplication The solution is. Seven times a number plus three times another number equals negative one. The sum of the two numbers is negative three. What are the numbers? Let x represent one number and y represent the second number. Notice that if you multiply the second equation by 3, the coefficients of the y terms are additive inverses. Now, substitute 2 for x in either equation to find y. So, the two numbers are 2 and 5. A field goal is 3 points and the extra point after a touchdown is 1 point. In a recent post season, Adam Vinatieri of the Indianapolis Colts made a total of 21 field goals and extra point kicks for 49 points. Find the number of field goals and extra points that he made. Let x represent the number of field goals and y represent the number of extra point kicks. Notice that if you multiply the first equation by 1, the coefficients of the y terms are additive inverses. Now, substitute 14 for x in either equation to find y. So, he made 14 field goals and 7 extra point kicks. Page 13

14 So, the two numbers are 2 and 5. A field goal is 3 points and the extra point after a touchdown is 1 point. In a recent post season, Adam Vinatieri of the Indianapolis Colts made a total of 21 field goals and extra point kicks for 49 points. Find the number of field goals and extra points that he made. Let x represent the number of field goals and y represent the number of extra point kicks. Notice that if you multiply the first equation by 1, the coefficients of the y terms are additive inverses. Now, substitute 14 for x in either equation to find y. So, he made 14 field goals and 7 extra point kicks. Use elimination to solve each system of equations. 2.2x + 3y = x + 2.1y = Notice that if you multiply the first equation by 2.1 and the second equation by 3, the coefficients of the y terms are additive inverses. Now, substitute 2.5 for x in either equation to find y. Page 14

15 So, he made 14 field goals and 7 extra point kicks. Use elimination to solve each system of equations. 2.2x + 3y = x + 2.1y = Notice that if you multiply the first equation by 2.1 and the second equation by 3, the coefficients of the y terms are additive inverses. Now, substitute 2.5 for x in either equation to find y. So, the solution is (2.5, 3.25). 0.4x y = x + y = 7.5 Notice that if you multiply the second equation by 0.25, the coefficients of the y terms are additive inverses. Now, substitute 4.5 for x in either equation to find y. So, the solution is (4.5, 1.5). Page 15

16 6-4 Elimination Using Multiplication So, the solution is (2.5, 3.25). 0.4x y = x + y = 7.5 Notice that if you multiply the second equation by 0.25, the coefficients of the y terms are additive inverses. Now, substitute 4.5 for x in either equation to find y. So, the solution is (4.5, 1.5). Notice that if you multiply the second equation by 8, the coefficients of the y terms are additive inverses. Now, substitute 3 for x in either equation to find y. Page 16

17 So, the solution is (4.5, 1.5). Notice that if you multiply the second equation by 8, the coefficients of the y terms are additive inverses. Now, substitute 3 for x in either equation to find y. So, the solution is. Notice that if you multiply the second equation by 12, the coefficients of the y terms are additive inverses. Page 17

18 6-4 Elimination Using So, the solution is Multiplication. Notice that if you multiply the second equation by 12, the coefficients of the y terms are additive inverses. Now, substitute So, the solution is x in either equation to find y.. CCSS MODELING -home nurses and support staff places necessary personal at locations on a daily basis. Each nurse placed works 240 minutes per day at a daily rate of $90. Each support staff employee works 360 minutes per day at a daily rate of $120. a. On a given day, 3000 total minutes are worked by the nurses and support staff that were placed. Write an b. On the same day, $1050 of total wages were earned by the placed nurses and support staff. Write an equation Solve the system of equations, and interpret the solution in context of the situation. Page 18 Let n represent the number of nurse s minutes and s represent the number of support staff minutes. 240n + 360s

19 So, the solution is. CCSS MODELING -home nurses and support staff places necessary personal at locations on a daily basis. Each nurse placed works 240 minutes per day at a daily rate of $90. Each support staff employee works 360 minutes per day at a daily rate of $120. a. On a given day, 3000 total minutes are worked by the nurses and support staff that were placed. Write an b. On the same day, $1050 of total wages were earned by the placed nurses and support staff. Write an equation Solve the system of equations, and interpret the solution in context of the situation. Let n represent the number of nurse s minutes and s represent the number of support staff minutes. 240n + 360s = n + 120s = 1050 Notice that if you multiply the second equation by 3, the coefficients of the s terms are additive inverses. Now, substitute 5 for n in either equation to find s. The graphs of x + 2y = 6 and 2x + y = 9 contain two of the sides of a triangle. A vertex of the triangle is at the intersection of the graphs. What are the coordinates of the vertex? Draw the graph of the two lines. Identify the vertex of the triangle. The line that forms the third side of the triangle is the line x y = 3. Draw this line on the previous graph. Name the other two vertices of the triangle. Notice that if you multiply the first equation by 2, the coefficients of the x terms are additive inverses. Page 19

20 Draw the graph of the two lines. Identify the vertex of the triangle. The line that forms the third side of the triangle is the line x y = 3. Draw this line on the previous graph. Name the other two vertices of the triangle. Notice that if you multiply the first equation by 2, the coefficients of the x terms are additive inverses. Now, substitute 1 for y in either equation to find x. The vertex is (4, 1). b. Rewrite each equation in slope-intercept form and then graph on the same coordinate plane. The vertex is (4, 1). c. Rewrite the equation in slope-intercept form and then graph the line on the same coordinate plane with the first two equations. Page 20

21 c. Rewrite the equation in slope-intercept form and then graph the line on the same coordinate plane with the first two equations. To find the other two vertices, use elimination to find the solutions to the other two systems of equations created by the three lines. Substitute 3 for y in either equation to find the value of x. So, the vertex is (0, 3). Substitute 2 for x in either equation to find the value of y. So, the vertex is (2, 5). At an entertainment center, two groups of people bought batting tokens and miniature golf games, as shown in the table. Define the variables, and write a system of linear equations from this situation. Page 21

22 So, the vertex is (2, 5). At an entertainment center, two groups of people bought batting tokens and miniature golf games, as shown in the table. Define the variables, and write a system of linear equations from this situation. Solve the system of equations, and explain what the solution represents. Let x = the cost of a batting token and let y = the cost of a miniature golf game; 16x + 3y = 30 and 22x + 5y = 43. b. Notice that if you multiply the first equation by 5 and the multiply the second equation by 3, the coefficients of the y terms are additive inverses. Now, substitute 1.5 for x in either equation to find y. The solution is (1.5, 2). A batting token costs $1.50 and a game of miniature golf costs $2.00. Mrs. Henderson discovered that she had accidentally reversed the digits of a test score and did not give a student 36 points. Mrs. Henderson told the student that the sum of the digits was 14 and agreed to give the student his correct score plus extra credit if he could determine his actual score. What was his correct score? Let x represent the tens digit and y represent the ones digit. The sum of the digits is 14, so x + y = 14. Since Mrs. Henderson switched the digits, she gave the students 10y + x points, when they actually earned 10x + y Page 22

23 The solution is (1.5, 2). A batting token costs $1.50 and a game of miniature golf costs $2.00. Mrs. Henderson discovered that she had accidentally reversed the digits of a test score and did not give a student 36 points. Mrs. Henderson told the student that the sum of the digits was 14 and agreed to give the student his correct score plus extra credit if he could determine his actual score. What was his correct score? Let x represent the tens digit and y represent the ones digit. The sum of the digits is 14, so x + y = 14. Since Mrs. Henderson switched the digits, she gave the students 10y + x points, when they actually earned 10x + y Notice that if you multiply the first equation by 9 the x terms are the same, so subtract the equations. Now, substitute 5 for y in either equation to find x. So, the correct score is 95. Explain how you could recognize a system of linear equations with infinitely many solutions. The system of equations 2x - 5y = 14 and 12x - 30y = 84 will have infinitely many solutions. You can solve by substitution or elimination and get a true statement, such as, 0 = 0. You can also notice that 12x - 30y = 84 is 6(2x 5y = 14). The system of equations will have infinitely many solutions whenever one of the equations is a multiple of the other. Jason and Daniela are solving a system of equations. Is either of them correct? Explain your reasoning. Page 23

24 The system of equations 2x - 5y = 14 and 12x - 30y = 84 will have infinitely many solutions. You can solve by substitution or elimination and get a true statement, such as, 0 = 0. You can also notice that 12x - 30y = 84 is 6(2x 5y = 14). TheUsing systemmultiplication of equations will have infinitely many solutions whenever one of the equations is a multiple of 6-4 Elimination the other. Jason and Daniela are solving a system of equations. Is either of them correct? Explain your reasoning. Jason is correct. In order to eliminate the r terms, you must multiply the second equation by 2 and then subtract, or multiply the equation by 2 and then add. When Daniela subtracted the equations, she should have gotten r + 16t = 18, instead of r = 18. The t-term should not be eliminated. She needs to find multiples of the equations that have the r or t before adding or subtracting the equations. Write a system of equations that can be solved by multiplying one equation by 3 and then adding the two equations together. Sample answer: 2x + 3y = 6, 4x + 9y = 5 Substitute for x in either equation and solve for y. Page 24

25 Jason is correct. In order to eliminate the r terms, you must multiply the second equation by 2 and then subtract, or multiply the equation by 2 and then add. When Daniela subtracted the equations, she should have gotten r + 16t = 18, instead ofusing r = 18.Multiplication The t-term should not be eliminated. She needs to find multiples of the equations that have the 6-4 Elimination r or t before adding or subtracting the equations. Write a system of equations that can be solved by multiplying one equation by 3 and then adding the two equations together. Sample answer: 2x + 3y = 6, 4x + 9y = 5 Substitute for x in either equation and solve for y. The solution of the system 4x + 5y = 2 and 6x Discuss the steps that you used. 2y = b is (3, a). Find the values of a and b. We already know that the value of x is 3. We can use that to find y. Substitute 3 for x and solve for y. So y = 2, which means that a = 2. Now substitute 2 for y and 3 for x into the second equations and solve for b. a = 2, b = 22 WRITING IN MATH Why is substitution sometimes more helpful than elimination, and vice versa? Page 25

26 The solution of the system 4x + 5y = 2 and 6x Discuss the steps that you used. 2y = b is (3, a). Find the values of a and b. We already know that the value of x is 3. We can use that to find y. Substitute 3 for x and solve for y. So y = 2, which means that a = 2. Now substitute 2 for y and 3 for x into the second equations and solve for b. a = 2, b = 22 WRITING IN MATH Why is substitution sometimes more helpful than elimination, and vice versa? Sample answer: It is more helpful to use substitution when one of the variables has a coefficient of 1 or if a coefficient can be reduced to 1 without turning other coefficients into fractions. Otherwise, elimination is more helpful because it will avoid the use of fractions when solving the system. What is the solution of this system of equations? (3, 3) ( 3, 3) ( 3, 1) (1, 3) Notice that if you multiply the second equation by 2, the coefficients of the x terms are additive inverses. Now, substitute 1 for y in either equation to find x. Page 26

27 Sample answer: It is more helpful to use substitution when one of the variables has a coefficient of 1 or if a coefficient can be reduced to 1 without turning other coefficients into fractions. Otherwise, elimination is more helpful because 6-4 Elimination Using Multiplication it will avoid the use of fractions when solving the system. What is the solution of this system of equations? (3, 3) ( 3, 3) ( 3, 1) (1, 3) Notice that if you multiply the second equation by 2, the coefficients of the x terms are additive inverses. Now, substitute 1 for y in either equation to find x. So, the solution is ( 3, 1) and the correct choice is C. A buffet has one price for adults and another for children. The Taylor family has two adults and three children, and their bill was $ The Wong family has three adults and one child. Their bill was $38. Which system of equations could be used to determine the price for an adult and for a child? If x is the adult ticket price and y is the child s ticket price, then the Taylor family s price would be 2 adults (2x) and 3 children (3y). That means that their price could be represented by. Thus, you can eliminate choices F and J. The Wong family has 3 adults (3x) and 1 child (1y), so their price could be represented by. Therefore, the correct choice is G. A customer at a paint store has ordered 3 gallons of ivy green paint. Melissa mixes the paint part yellow. How many quarts of blue paint does she use? Page 27 in a Manual ratio of- Powered 3 parts by blue to one esolutions Cognero Let b represent the number of quarts of blue paint mixed and y represent the number of quarts of yellow paint

28 If x is the adult ticket price and y is the child s ticket price, then the Taylor family s price would be 2 adults (2x) and 3 children (3y). That means that their price could be represented by. Thus, you can eliminate choices F and J. The Wong family has 3 adults (3x) and 1 child (1y), so their price could be represented by. Therefore, the correct choice is G. A customer at a paint store has ordered 3 gallons of ivy green paint. Melissa mixes the paint in a ratio of 3 parts blue to one part yellow. How many quarts of blue paint does she use? Let b represent the number of quarts of blue paint mixed and y represent the number of quarts of yellow paint equations that describes the problem. first equation: + y = 12 If the first equation is solved for y, you get y = 12 - b. Substitute this into the second equation and solve for b. Therefore, she will need 9 quarts of blue paint. The table shows the results of a number cube being rolled. What is the experimental probability of rolling a 3? Outcome Frequency Find the total number of outcomes = = 20. The experimental probability is. Page 28

29 Therefore, she will need 9 quarts of blue paint. The table shows the results of a number cube being rolled. What is the experimental probability of rolling a 3? Outcome Frequency Find the total number of outcomes = = 20. The experimental probability is. So, the correct choice is D. Use elimination to solve each system of equations. f +g= 3 f g=1 Notice the coefficients for the g terms are the opposite, so add the equations. Now, substitute 1 for f in either equation to find g. The solution is ( 1, 2). Page 29

30 Find the total number of outcomes = = 20. The experimental probability is. So, the correct choice is D. Use elimination to solve each system of equations. f +g= 3 f g=1 Notice the coefficients for the g terms are the opposite, so add the equations. Now, substitute 1 for f in either equation to find g. The solution is ( 1, 2). 6g + h = 7 6g + 3h = 9 Notice the coefficients for the g terms are the same, multiply equation 2 by 1, then add the equations to find h. Now, substitute 1 for h in either equation to find g. Page 30

31 6-4 Elimination Using The solution is ( 1, Multiplication 2). 6g + h = 7 6g + 3h = 9 Notice the coefficients for the g terms are the same, multiply equation 2 by 1, then add the equations to find h. Now, substitute 1 for h in either equation to find g. The solution is ( 1, 1). 5j + 3k = 9 3j + 3k = 3 Notice the coefficients for the k terms are the same, multiply equation 2 by 1 and add the equations to find j. Now, substitute 3 for j in either equation to find k. Page 31

32 The solution is ( 1, 1). 5j + 3k = 9 3j + 3k = 3 Notice the coefficients for the k terms are the same, multiply equation 2 by 1 and add the equations to find j. Now, substitute 3 for j in either equation to find k. The solution is ( 3, 2). 2x 4z = 6 x 4z = 3 Notice the coefficients for the z terms are the same, so multiply equation 2 by 1 and add the equations to find x. Now, substitute 9 for x in either equation to find z. Page 32

33 The solution is ( 3, 2). 2x 4z = 6 x 4z = 3 Notice the coefficients for the z terms are the same, so multiply equation 2 by 1 and add the equations to find x. Now, substitute 9 for x in either equation to find z. The solution is (9, 3). 5c 3v = 9 5c + 2v = 6 Notice the coefficients for the c terms are the opposite, so add the equations. Now, substitute 3 for v in either equation to find c. Page 33

34 The solution is (9, 3). 5c 3v = 9 5c + 2v = 6 Notice the coefficients for the c terms are the opposite, so add the equations. Now, substitute 3 for v in either equation to find c. The solution is (0, 3). 4b 3b 6n = 36 6n = 36 Notice the coefficients for the n terms are the same, so multiply equation 2 by 1 and add the equations to find b. Now, substitute 0 for b in either equation to find n. Page 34

35 The solution is (0, 3). 4b 3b 6n = 36 6n = 36 Notice the coefficients for the n terms are the same, so multiply equation 2 by 1 and add the equations to find b. Now, substitute 0 for b in either equation to find n. The solution is (0, 6). Brandy and Adriana work at an after school child care center. Together they cared for 32 children this week. Brandy cared for 0.6 times as many children as Adriana. How many children did each girl care for? Let b represent the number kids Brandy watched and a represent the number of kids Adriana watched. b + a = 32 b = 0.6a Substitute 0.6a for b in the first equation to find the value of a. Now, substitute 20 for a in either equation to find b. So, Brandy cared for 12 children and Adriana cared for 20 children. Page 35

36 The solution is (0, 6). Brandy and Adriana work at an after school child care center. Together they cared for 32 children this week. Brandy cared for 0.6 times as many children as Adriana. How many children did each girl care for? Let b represent the number kids Brandy watched and a represent the number of kids Adriana watched. b + a = 32 b = 0.6a Substitute 0.6a for b in the first equation to find the value of a. Now, substitute 20 for a in either equation to find b. So, Brandy cared for 12 children and Adriana cared for 20 children. Solve each inequality. Then graph the solution set. m 5 and The solution set is {m m m To graph the solution set, graph m 3}. and graph m 3. Then find the intersection. q + 11 < 5 and The solution set is {q q < 6 and q > 16}. To graph the solution set, graph q < 6 and graph q > 16. Then find the intersection. 2w + 9 > 11 or Page 36

37 The solution set is {q q < 6 and q > 16}. To graph the solution set, graph q < 6 and graph q > 16. Then find the intersection. 2w + 9 > 11 or The solution set is {w w > 1 or w < 10}. Notice that the graphs do not intersect. To graph the solution set, graph w > 1 and graph w < 10. Then find the union. 2r + 1 or The solution set is {r r or r 5}. Notice that the graphs do not intersect. To graph the solution set, graph r and graph r 5. Then find the union. Translate each sentence into a formula. The area A of a triangle equals one half times the base b times the height h. Rewrite the verbal sentence so it is easier to translate. The area A of a triangle equals one half times the base b times the height h. A equals times b times h A = b h The circumference C of a circle equals the product of 2,, and the radius r. Rewrite the verbal sentence so it is easier to translate. The circumference C of a circle equals the product of 2,, and the radius r. C equals 2 times times r C = 2 r The volume V of a rectangular box is the length times the width w multiplied by the height h. Page 37

38 = A b h The circumference C of a circle equals the product of 2,, and the radius r. Rewrite the verbal sentence so it is easier to translate. The circumference C of a circle equals the product of 2,, and the radius r. C equals 2 times times r C = 2 r The volume V of a rectangular box is the length times the width w multiplied by the height h. Rewrite the verbal sentence so it is easier to translate. The volume V of a rectangular box is the length the width w multiplied by the height h. V equals V = times w times w h h The volume of a cylinder V is the same as the product of height h. and the radius r to the second power multiplied by the Rewrite the verbal sentence so it is easier to translate. The volume V of a cylinder is the same as the product of and the radius r to the second power multiplied by the height h. V equals times r squared times h 2 V = r h The area of a circle A equals the product of and the radius r squared. Rewrite the verbal sentence so it is easier to translate. The area of a circle A equals the product of radius r squared. A equals times r squared 2 A = r and the Acceleration A equals the increase in speed s divided by time t in seconds. Rewrite the verbal sentence so it is easier to translate. Acceleration A equals speed s divided by time t in seconds. A equals s divided by t A = s t Page 38