Unit 4: Systems of Linear Equations &...

Intermediate Algebra Unit 4: Systems of Linear

Equations & Inequalities

Intermediate Algebra Unit 4: Systems of Equations & Inequalities

Objectives: page

graph a line given its equation 2 – 3

solve systems of linear equations graphically 4 – 6

solve systems of linear equation algebraically using substitution method 7 – 11

solve systems of linear equation algebraically using elimination method 12 – 16

systems of linear equations review questions 17 – 24

solving real-world application questions using systems of equations 25 – 30

solving systems of inequalities by graphing 31 – 34

linear programming 35 – 37

linear programming applications 38 – 42

Solving Systems of Linear Equations Graphically: For each system of equations, graph the system, solve the system, and check your solution. 1. 2y + x = 1 y + 2x = 5 2. y = -x y = 2x – 6

For each system of equations, graph the system, solve the system, and check your solution. 3. x + y = 2 2y – x = 10 4. y = x – 4 3y = x – 6

For each system of equations, graph the system, solve the system, and check your solution. 5. y – x = 2 y = 2x – 1 6. x – 2y = 14 x + 3y = 9

Do Now:

Substitution Method:

Solving Systems of Linear Equations Algebraically: Substitution Method 1. 4x + 3y = 27

y = 2x – 1 2. 3x – 4y = 26

x + 2y = 2

3. y = x + 1 x + y = 9

4. x = 3y + 1

5y – 2x = 1

5. x + y = 11 3x – 2y = 8

6. 3x – 2y = 11

x + 2y = 9

7. x – 2y = -2 2x – y = 5

8. x + y = 6

-2x + y = -6

Do Now:

Elimination Method:

Examples:

Solving Systems of Linear Equations Algebraically: Elimination Method 1. x – 2y = -4

-2x + 2y = 8 2. x + y = 6

-2x + y = -6

3. 5x – 2y = 22 x + 2y = 2

4. x + 2y = 7

x – y = -8

5. x + y = 7 2x + 3y = 21

6. 5x + 3y = 17

4x – 5y = 21

Solving Systems of Linear Equations: Mixed Practice Solve each system of equations using the indicated method and check your solutions:

(on the graph grids below) (on workspace pages 18 – 19) GRAPHICALLY ELIMINATION SUBSTITUTION

(1) y = -2x + 7 y = 3x – 3

(2) 3x + 2y = 4 -3x + 6y = -12

(3) x = 2y + 1 3x – y = 8

(4) 2y – 6 = 3x 4y + 3x = -24

(5) 6x + 2y = 36 2x – 8y = -14

(6) y = 2x + 1 x + y = 10

(1) (4)

(workspace for Mixed Practice)

Review Questions:

For the following system of equations:

• solve 3 graphically (use graph grids on workspace pages 21 – 22)

• solve 3 using elimination method (on workspace pages 23 – 24)

• solve 3 using substitution method (on workspace pages 23 – 24)

and check all of your solutions.

(1) y = - ⅔ x + 3 y = -2x – 1

(2) 3x + 5y = -16 2x + 3y = -9

(3) x – 3y = -27 3x + 2y = -4

(4) 2x – 7y = 49 x + 4y = 2

(5) x = 3y + 4 2x + 4y = 38

(6) x – 3y = -8 3x + 2y = 9

(7) x + 2y = 4 3x – 4y = 7

(8) y = - ¾ x + 2 2y = x – 6

(9) y = ⅔ x + 1 y = -2x – 3

(workspace for Review Questions)

(workspace for previous page)

Using Systems of Equations to Solve Word Problems:

SHOW ALL WORK to solve each of the following algebraically: (1) Conrad’s Balloon Bouquets charges $20 for balloon arrangements and delivery is $3 per

mile. The Balloon House charges $30 for balloon arrangements and delivery is $2 per mile. At what distance do the two stores charge the same amount for a balloon arrangement?

(2) To develop a roll of film, The Photo Lab charges $3.20 per roll plus 8 cents per print, and Specialty Photos charges $2.60 per roll plus 10 cents per print. (a) Write a system of equations that represent the cost of developing a roll of film at each lab. (b) Under what conditions is the cost to develop a roll of film the same for either store? (c) When is it best to use The Photo Lab and when is it best to use Specialty Photos?

(3) All 28 members in Crestview High School’s Ski Club went on a one-day ski trip. Members can rent skis for $16 per day or snowboards for $19 per day. The club paid a total of $478 for rental equipment. (a) Write a system of equations that represents the number of members who rented the two

types of equipment. (b) How many members rented skis and how many rented snowboards?

Systems Word Problems:

Define variables, write a system of equations, and solve algebraically to answer the following questions. SHOW ALL WORK!

(1) Christina and Marlena opened their first savings accounts on the same day. Christina

opened her account with $50 and plans to deposit $10 every month. Marlena opened her account with $30 and plans to deposit $15 every month. After how many months will their two accounts have the same amount of money? What will that amount be?

(2) It takes Akira 10 minutes to make a black and white drawing and 25 minutes for a color drawing. On Saturday he made a total of 9 drawings in 2 hours. Write and solve a system of equation to determine how many drawings of each type Akira made.

(3) Cara and Gabriella are selling pies for a school fundraiser. Customers can buy apple pies and lemon meringue pies. Cara sold 6 apple pies and 4 lemon meringue pies for a total of $80. Gabriella sold 6 apple pies and 5 lemon meringue pies for a total of $94. What is the cost each of one apple pie and one lemon meringue pie?

(4) Sam spent $24.75 to buy 12 flowers for his mother. The bouquet contained roses and daisies. Roses cost $2.50 each and daisies cost $1.75 each. How many of each type of flower did Sam buy?

(5) Marc's school is selling tickets to the annual dance competition. On the first day of ticket sales the school sold 3 senior citizen tickets and 5 child tickets for a total of $70. The school took in $216 on the second day by selling 12 senior citizen tickets and 12 child tickets. Find the price of a senior citizen ticket and the price of a child ticket.

Solving Systems of Inequalities by Graphing:

Solving Systems of Inequalities by Graphing: 1. y ≤ 7

2x – y ≤ 3 x + 2y ≥ -6

2. x ≤ -1 y ≤ 3x + 2 y ≥ -3x – 10

3. y ≥ x

y ≤ x + 6 x ≤ 6 x ≥ -2

4. x ≥ -2 x ≤ 3 y ≥ -x + 1 y ≤ 4

5. x ≥ 0 -1/3 x + y ≥ -4 1/3 x + y ≤ -1

6. y ≤ x + 6 y ≥ x + 1 y ≤ -x + 6 y ≥ -x – 1

7. y ≤ x

y ≤ -x + 2 y ≥ 0

8. y ≤ -x + 8 y ≤ 2x + 2 y ≥ 1/2 x – 4 y ≥ -5/2 x + 2

Do Now: (1) Graph the following system of inequalities:

x ≤ 5

y ≥ -3x

2y ≤ x + 7

y ≥ x – 4

(2) Name the coordinates of the vertices of the feasible region. (3) Find the maximum and minimum values of the given function for this region.

Introduction to Linear Programming:

When solving problems with inequalities, the boundary lines are the for the situation or conditions given to the variables.

The intersection of the graphs of the system of inequalities is the .

When the graph of constraints creates a polygon, the region is (although, it is possible that a system of inequalities forms an region, where part of it is open).

The maximum or minimum value of a related function always occurs at one of the of the feasible region.

Linear Programming: (4) Graph the following constraints and shade the feasible region:

x ≥ 0

y ≥ 0

y – 3 ≤ 3x

y + x ≤ 7

(5) Name the coordinates of the vertices of the feasible region. (6) Using the function P = 4x + 3y, find the maximum and minimum values for this region.

Linear Programming: Graph each of the following constraints and shade the feasible region:

(1) x ≥ 0 y ≥ 0

y ≤ 23 x +1

y ≤ -x + 6

(2) x ≥ 0 y ≥ 0 y ≥ 2x +1 y ≤ -2x + 9

Name the coordinates of the vertices of the feasible region:

Find the maximum and minimum values for each region using the given function:

P = 2x + 5y

P = 3x + 6y

Linear Programming Applications: (1) Yum’s Bakery bakes two types of breads, A and B. One batch of bread A uses 5 pounds

of oats and 3 pounds of flour. One batch of bread B uses 2 pounds of oats and 3 pounds of flour. The company currently has 180 pounds of oats and 135 pounds of flour available. One batch of bread A yields $40 of profit, and one batch of bread B yields $30 of profit. If Yum’s Bakery wants to maximize its profits, how many batches of each type of bread should they bake?

(2) Baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flour. A tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour. A baker has 16 cups of milk and 15 cups of wheat flour. He makes $3 profit per tray of corn muffins and $2 profit per tray of bran muffins. How many trays of each type of muffin should the baker make to maximize his profit?

(3) Trees in urban areas help keep air fresh by absorbing carbon dioxide. Exponential City has a total of $2100 to spend on planting spruce and maple trees. The total land available for planting is 45,000 ft2. Use the given table of information to determine how many of each tree should the city plant to maximize carbon dioxide absorption.

Spruce Maple

Planting Cost $30 $40 Area Required 600 ft2 900 ft2 Carbon Dioxide

Absorption 650 lbs/year 300 lbs/year

(4) Ace Guitars produces acoustic and electric guitars. Each acoustic guitar yields a profit of $30, and requires 2 work hours in factory A and 4 work hours in factory B. Each electric guitar yields a profit of $50 and requires 4 work hours in factory A and 3 work hours in factory B. Each factory operates for at most 10 hours per day. Find the number of each type of guitar that should be produced each day to maximize the company’s profits.

(notes & work space for packet)

Unit 4: Systems of Linear Equations &...

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