unit 3 Systems

Standard Topic 16a Solve a system of equations by substitution & graphing 16b Solve a system of equations by elimination 17a Solve Coin Problems 17b Solve RTD Problems 17c Solve Mixture Problems 17d Solve Other System Word Problems 18 Solve a System of Inequalities by graphing

3.1 Systems of Equations by Graphing and Substitution Review

Horizontal and Vertical Lines (only one variable in the equation)

I can solve a system of equations by graphing or

by substitution.

H O Y

V

U

x

y=-3

x=5

Solution: ( , )

x=4 y=2

x=-3 y=-5

Solve By Graphing −𝑥 + 2𝑦 = −6 2𝑥 + 3𝑦 = −9  3𝑥 + 2𝑦 = 2 𝑥 = 3  

𝑦 = !

!𝑥 + 5 𝑦 = !

!𝑥 − 4

𝑥 = −3 𝑦 = −𝑥 + 3

Solve By Substitution −𝑥 + 9𝑦 = −5 𝑦 = !

!𝑥 − 1

𝑥 = 5𝑦 + 1 𝑦 = 3 7𝑥 + 4𝑦 = 24 𝑦 = !

!𝑥 − 5

4𝑥 = 16 𝑦 = − !!𝑥 + 7

3.1 Systems of Equations by Graphing and Substitution Review Practice 1. 2𝑥 − 𝑦 = 1

𝑦 = −3 2. 𝑥 = 12𝑥 + 𝑦 = 4 3. 3𝑥 + 𝑦 = −3

3𝑥 + 𝑦 = 3

4. 𝑦 = 𝑥 + 2𝑥 − 𝑦 = −2 5. 𝑥 + 3𝑦 = −3

𝑥 − 3𝑦 = −3 6. 𝑦 − 𝑥 = −1𝑥 + 𝑦 = 3

7. 𝑥 − 𝑦 = 3𝑥 − 2𝑦 = 3 8.

𝑥 + 2𝑦 = 4𝑦 = !

!𝑥 + 2

9. Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $20 in operating costs per week plus $0.50 to produce each treat. He plans to sell each treat for $1.50. Graph the system of equations 𝑦 = 0.5𝑥 + 20 and 𝑦 = 1.5𝑥 to represent the situation. How many treats does Nick need to sell per week to break even? Solve the system of equations by substitution.

10. 𝑦 = 2𝑥 − 2

𝑦 = 𝑥 + 2 11. 𝑦 = 2𝑥 + 62𝑥 − 𝑦 = 2 12. 3𝑥 + 𝑦 = 12

𝑦 = −𝑥 − 2

13. 𝑥 = 13− 2𝑦−2𝑥 − 3𝑦 = −18 14. 𝑥 = 3+ 2𝑦

4𝑥 − 8𝑦 = 12 15. 𝑥 = 36+ 5𝑦2𝑥 + 𝑦 = −16

16. 2𝑥 − 3𝑦 = −24𝑥 = 18− 6𝑦 17. 𝑥 = 84− 14𝑦

2𝑥 − 7𝑦 = −7 18. 0.3𝑥 = 0.2𝑦 = 0.5𝑥 = 2𝑦 − 5

19. Kenisha sells athletic shoes part-time at a department store. She can earn either $500 per month plus a 4% commission on her total sales, or $400 per month plus a 5% commission on total sales. Write a system of equations to represent the situation. What is the total price of the athletic shoes Kenisha needs to sell to ear the same income from each pay scale? Which is the better offer?

Review Questions (Complete ALL) 20. Solve -21 < 2x +1 < 6 23. If h(x) = x2 – 5 what is the value of h(-7)? 24. Given the sequence 53, 48, 43, 38… Write an equation for the sequence to find the nth term. Find the 16th term. 25. Write the inequality for the graph.

21. Graph the equation 3y + 2x = 9

3.2 Systems of Equations by Elimination

Solve By Elimination 1. Both equations should be in STANDARD FORM: Ax +By = C 2. Either A or B should be the same #, but opposite signs. 𝑥 + 2𝑦 = 6 𝑥 − 2𝑦 = −2 3. Add DOWN to eliminate. 𝑥 + 2𝑦 = 6 𝑥 − 2𝑦 = −2 4. Substitute answer back into ONE of the equations (you pick which one) If A or B are NOT the same numbers, Make them the same. 2𝑥 + 𝑦 = 233𝑥 + 2𝑦 = 37 4𝑥 + 2𝑦 = 8

3𝑥 + 3𝑦 = 9

I can solve a system of

equations by elimination.

3.2 Systems of Equations by Elimination Practice 1. 𝑥 − 𝑦 = 1𝑥 + 𝑦 = −9 2. 𝑝 + 𝑞 = −2

𝑝 − 𝑞 = 8 3. 4𝑥 + 𝑦 = 233𝑥 − 𝑦 = 12

4. 2𝑥 + 5𝑦 = −32𝑥 + 2𝑦 = −14 5. 3𝑥 + 2𝑦 = −1

4𝑥 + 2𝑦 = −6 6. 7𝑥 + 2𝑦 = 27𝑥 − 2𝑦 = −30

7. 2𝑥 − 𝑦 = −13𝑥 − 2𝑦 = 1 8. 5𝑥 − 2𝑦 = −10

3𝑥 + 6𝑦 = 66 9. 7𝑥 + 4𝑦 = −45𝑥 + 8𝑦 = 28

10. 3𝑥 + 4𝑦 = 275𝑥 − 3𝑦 = 16 11. 6𝑥 + 3𝑦 = 21

2𝑥 + 2𝑦 = 22 12. −3𝑥 + 2𝑦 = −152𝑥 − 4𝑦 = 26

Review Questions (Complete ALL) 13. As the number of farms has decreased in the United States, the average size of the remaining farms has grown larger, as shown in the table below. Write the linear regression equation? What does the slope mean in this context? What does the y-intercept mean in this context? Calculate the correlation coefficient and interpret its meaning. 14. Solve: 8(x - 2) – 4x + 4x = 22 15. Solve 3𝑥 − 7 = 11 16. The football team had 84 players the first week of practice. Due to all the conditioning the team lost 3 players per week. Write an equation to represent the situation. 17. Is the following relation a function? {(4,12), (5, 16), (7, 19), (10, 22)} 18. Write an equation to find the nth term for the following sequence: -7, -2, 3, 8…

3.3 Systems Word Problems (RTD) & (RTW) One distance = Other distance Distance one + Distance two = Total

Rate∙Time = Distance Melanie left the city traveling at 58 mph, while, at the same time, Jessica left the city going the opposite direction at a speed of 51

mph. Find the time Melanie traveled before the two were 199 miles apart. Sketch distance example:

John left home and drove at the rate of 45 mph for 20 hours. He stopped for lunch then drove for another 3 hours at the rate of 55 mph to reach his destination. How many miles did John drive?

Sketch distance example:

Rate Time Distance

Rate Time Distance

Example 2

Example 1

I can solve a system of

equations RTD & RTW word problems.

A train leaves a train station at 1 p.m. It travels at an average rate of 60 mi/h. A high-speed train leaves the same station an hour later. It

travels at an average rate of 96 mi/h. The second train follows the same route as the first train on a track parallel to the first. In how many hours will the second train catch up with the first train? Sketch distance example:

Noya drives into the city to buy a software program at a computer store. Because of traffic conditions, she averages only 15 mi/h. On

her drive home she averages 35 mi/h. If the total travel time is 2 hours, how long does it take her to drive to the computer store? Sketch distance example:

Rate∙Time = Work Suppose one painter can paint the entire house in twelve hours, and

the second painter takes eight hours. How long would it take the two painters together to paint the house?

Rate Time Distance

Rate Time Distance

Rate Time Work

Example 3

Example 4

Example 5

Suppose that it takes Janet 6 hours to paint her room if she works alone and it takes Carol 4 hours to paint the same room if she works

alone. How long will it take them to paint the room if they work together?

One garden hose can fill an above-ground pool in 10 hours. A second hose can fill the pool twice as fast as the first one. If both

hoses are used together to fill the pool, how many hours will it take?

Rate Time Work

Rate Time Work

Example 6

Example 7

3.3 Systems Word Problems (RTD) & (RTW) Practice 1. A car and a bus set out at 2 p.m. from the same point, headed in the same

direction. The average speed of the car is 30 mph slower than twice the speed of the bus. In two hours, the car is 20 miles ahead of the bus. Find the rate of the car.

2. Noya drives into the city to buy a software program at a computer store. Because

of traffic conditions, she averages only 15 mi/h. On her drive home she averages 35 mi/h. If the total travel time is 2 hours, how long does it take her to drive to the computer store?

3. Jane and Peter leave their home traveling in opposite directions on a straight

road. Peter drives 15 mi/h faster than Jane. After 3 hours, they are 225 miles apart. Find Peter's rate and Jane's rate.

4. Two bicyclists ride in opposite directions. The speed of the first bicyclist is 5 miles

per hour faster than the second. After 2 hours they are 70 miles apart. Find their rates.

5. A Speedy River barge bound for New Orleans leaves Baton Rouge, Louisiana, at 9:00 A.M. and travels at a speed of 10 miles per hour. A Rail Transport freight train also bound for New Orleans leaves Baton Rouge at 1:30 P.M. the same day. The train travels at 25 miles per hour, and the river barge travels at 10 miles per hour. Both the barge and the train will travel 100 miles to reach New Orleans.

a. How far will the train travel before catching up to the barge?

b. Which shipment will reach New Orleans first? At what time?

c. If both shipments take an hour to unload before heading back to Baton Rouge, what is the earliest time that either one of the companies can begin to load grain to ship to Baton Rouge?

6. Benny can dust the house in 11 hours. Mary can dust the same house in 7 hours.

How long would it take them to dust the house together? 7. Benny and Tim were able to cut the shrubs in 7 hours together. It takes Tim 11 hours

to complete the same job along. Without help, how long would it take Benny to complete the same job?

Review Questions (Complete ALL) 8. The football team had 84 players the first week of practice. Due to all the conditioning the team lost 3 players per week. Write an equation to represent the situation. 9. Solve 10𝑥 + 2 = −48 10. 11. 12. Solve the systems of equations

2𝑥 + 7𝑦 = 1𝑥 + 5𝑦 = 2

13. Solve −15 ≤ 6𝑥 − 3 < 10

Write the inequality for the graph.

Write the equation of the line.

14. Write the equation of the line from the table.

X Y

0 -24

3 -18

6 -12

9 -6

3.4 Systems Word Problems (Coin & Population)

Coin Problems Rocco has 28 coins all of which are either dimes or nickels. The coins are worth a total of $2.60. Find how many dimes and how many nickels he has.

Rosa has $3.10 in nickels and dimes. She has 10 less nickels than she has dimes. How many nickels and dimes does she have?

A bank teller has a total of 124 bills in fives and tens. The total value of the money is $840. How many of each kind does he have?

Vinnie has $3.60 in coins, all of which are quarter or dimes. There are 27 coins. Find out how many quarter and how many dimes Vinnie has.

Dimes Nickels Total # of Coins $ Amount

Dimes Nickels Total # of Coins $ Amount

Fives Tens Total # of Bills $ Amount

Quarters Dimes Total # of Coins $ Amount

I can solve a system of

equations Coin & population word

problems.

Population Problems Central High School is increasing in enrollment by 25 students each year, Park Hill High School is decreasing by 12 students each year. If Park Hill has 2956 students & Central has 1989 students, how many years will it take until they have the same enrollment? In a recent poll of 30,000 voters, Donald Trump was leading with 9,500 votes and Carly Fiorina had 6,744 votes. After the debate, Trump was losing 12 votes each day and Fiorina was gaining 17 votes each day. After how many days will each candidate have the same amount of votes?

Other Problems

Tickets to the local school play cost $5.00 for adults and $2.00 for children. For the one night show, 285 tickets were sold and $1065 was collected. How many of each type of ticket were sold?

The sum of two numbers is 75. The larger number is 3 less than twice the smaller. Find the numbers.

The units digit of a two digit number is 1 less than three times the tens digit. The sum of the digits is 11. Find the original number.

Adults Children Total Tickets $ Amount

3.4 Systems Word Problems (Coin & Population) Practice

1. A collection of quarters and nickels is worth $1.25. There are 13 coins in all. How many of each are there?

2. A collection of nickels and dimes is worth $2.90. There are 19 more nickels than dimes. How many of each are there?

3. Tia and Ken each sold snack bars and magazine subscriptions for a school fund-raiser, as shown in the table. Tia earned $132 and Ken earned $190. What was the price per snack bar?

4. Michael has $1.95 total in his collection, consisting of quarter and nickels. The number of nickels is three more than the number of quarters. How many nickels and how many quarters does Michael have?

5. A library contains 2000 books. There are 3 times as many non-fiction books as fiction books. Determine how many fiction and non-fiction books are in the library.

6. Mac’s wallet is full of $5 and $10 bills. He has 25 bills totaling $230. How many of each bill does he have?

7. Jack has a collection of new nickels and quarters. He has a total of 50 coins worth $10.30. How many of each coin does he have?

8. The sum of two numbers is 24. Their difference is 15.

9. What are the two numbers? A large McDonald’s chocolate milkshake has 720 more calories than a double cheeseburger. Two double cheeseburgers and the large chocolate shake have a total of 2040 calories. How many calories are in each item?

10. The school that Stefan goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Find the price of a senior citizen ticket and the price of a child ticket.

11. At an ice cream parlor, ice cream cones cost $1.10 and sundaes cost $2.35. One day, the receipts for a total of 172 cones and sundaes were $294.20. How many cones were sold?

Review Questions (Complete ALL) 12. Solve the system of equations: 13. 𝑦 = −4𝑥 + 11

3𝑥 + 𝑦 = 9

14.

15. Solve 𝑛 + 7 − 2 = 12 16. Solve − !

!𝑥 = 4

17. Solve the system by graphing. y = -3 x = 6

Mara and Ling each recycled aluminum cans and newspapers, as shown in the table. Mara earned $3.77, and Ling earned $4.65. What was the price per pound of aluminum?

The figures have the same perimeter. Solve for the value of x.

3.5 Systems of Inequalities

Solving a System of Inequalities

𝑦 > −2𝑥 + 2𝑦 − 𝑥 ≤ 1

More examples: 𝑦 > −1𝑥 < 0 𝑦 < 𝑥 + 2

3𝑥 + 4𝑦 ≥ 12 2𝑥 + 𝑦 > 1𝑥 − 𝑦 ≥ −2

I can solve a linear inequality and use the graph to find the solution to a system of inequalities.

3.5 Systems of Inequalities Graph each inequality. 1. 𝑦 > 𝑥 − 2 2. 𝑦 ≥ 𝑥 + 2 3. 𝑥 + 2𝑦 > 1 Solve each system of inequalities by graphing. 4. 𝑦 > 𝑥 − 2

𝑦 ≤ 𝑥 5. 𝑦 > 𝑥 − 42𝑥 + 𝑦 ≤ 2 6. 𝑦 < 2𝑥 + 4

𝑦 ≥ 𝑥 + 1

7. 2𝑥 − 𝑦 ≥ 2𝑥 − 2𝑦 ≥ 2 8. 𝑥 + 𝑦 ≥ 1

𝑥 + 2𝑦 > 1 9. 𝑦 ≥ 𝑥 + 2𝑦 > 2𝑥 + 3

11. Emily wants to buy turquoise stones on her trip to New Mexico to give to at least 4 of her friends. The gift shop sells stones for either $4 or $6 per stone. Emily has no more than $30 to spend. Make a graph showing the number of each price of stone Emily can purchase. List three possible solutions. Tell whether the ordered pair is a solution of the system of inequalities. 12. (3,0) 13. (2,2) 14. (-2,2)

15. (1, -1) 16. (0, -1) 17. (1, 4)

Review Questions (Complete ALL) 18. 19. The sum of two consecutive

integers is 99. What are the integers?

20. 21. 22. Solve the system. 23. y = -2x +11 y +2x = 23 24. Write the equation of the line.

25.

Graph -2x+3y=21

Is the graph a function? Explain how you know.

Which equation best models the relationship between x and y values in the table?

a. y  =  x-­‐5  b. y  =  2x-­‐5  c. y  =  3x-­‐7  d. y  =  4x-­‐7  

24. Tandy cleaned out the change in her car and had 27 quarters and nickels that totaled $3.35. Use a system of equations to find out how many of each coin she had.

Write the inequality for the graph.