Unit 6: Systems of Equations Date

Math: Algebra 2B Name: Unit 6: Systems of Equations Date:

Day 1 –Solving By Graphing

 Two or more linear equations form a system of linear equations. Example: y = 2x – 1 y = 4x – 7  Any ordered pair that makes both equations true in the system is called the solution. The solution to the above system is: (3, 5). This is the point where the two lines intersect.

Three Types of Systems

Type Number of Solutions & Example Visual Description Representation I. Exactly one solution. y = 2x – 1 Consistent The lines have different slopes. y = 4x – 7 Independen t II. Infinitely many solutions. -6y = 4x – 6 Consistent The lines have ______Dependent slopes and ______y-intercepts. 2 y= - x +1 3 III. No solutions. The lines have the y = 3x – 1 Inconsistent ______with different ______. y = 3x + 2 They are ______. Steps for solving graphically:

1) Solve each equation “for y” if needed.

2) Graph and label each line.

3) Find the point where the lines intersect. Label that coordinate on your graph.

4) Check the solution algebraically or graphically using your calculator.

Example 1: a.) Circle: consistent independent / consistent dependent / inconsistent.

b.) Number of solutions: ______

1 c.) Solve the following system graphically.

d.) Check your answer algebraically if possible. y = ½x - 2 y = -3x + 5

To check the previous problem graphically in your calculator, here are the steps:

Another way to find the solution is to read it from the table: the y columns must match

Solution to the system: (2, -1)

Example 2:

a.) Circle Type: consistent independent / consistent dependent / inconsistent.

c.) Solve the following system graphically

d.) Check your answer graphically if possible.

-y = - 6x + 7 18x -3y = 21

Example 3:

a.) Circle Type: consistent independent / consistent dependent / inconsistent.

c.) Solve the following system graphically

2 d.) Check your answer if possible.

y = 6x + 4 5 + y = 6x

Example 4: Scientists studied the weights of two alligators over a period of 12 months. The initial weight and growth rate of each alligator are shown below. After how many months did the alligators weigh the same amount?

Let Statements:

Equations: Alligator 1:

Alligator 2:

Practice: Solve each system & check your answer using any method discussed in class. 5) y = 3 6) y = 8, x = -2 7) y = 3x – 3 3y = 2x - 9 -3y = -9x + 9

3 8) y = -3x + 2 9) y + 2x – 9 = 0 10) Thinking Question: 3x + y = 6 x + y – 1 = 0 One equation in a system is y = ½ x – 2.

a) Write a second equation so that the system has one solution. ______

b) Write a second equation so that the system has no solution. ______

c) Write a second equation so that the system has infinitely many solutions. ______

4 Day 2 – Applications of Systems by Graphing

WARM - UP

1. Match each type of system with the number of solutions the system has.

A. inconsistent i. exactly one

B. consistent and dependent ii. infinitely many

C. consistent and independent iii. no solution

2. Find the value of two numbers that have a sum of 12 and a difference of 2.

a.) Create two equations to represent the situation then solve each of them for y.

b.) Graph the 2 lines and find their solution.

Example 1: Suppose one puppy weighs 5 pounds at birth and grows at a rate of .25 lbs. per month over the first several months. Another puppy weighs 4 lbs. at birth and grows at a rate of .5 lbs. per month. After how many months will the puppies weigh the same amount? Use a chart, create an equation, or graph the situation.

5 Solving Graphically Using Your Calculator

*Note: In the notes, the 5 box model is provided for you, on the homework, you have to create your own 5 boxes to show your work.

Example 2: You need an electrician to do work on your home. You record the information about the electricians in the table below. How long would the job take if the total cost for each electrician is the same?

Equations Window Picture Answers & Buttons Pressed

Example 3: Moolah has 27 coins that are all dimes and quarters. The value of the coins is $4.35. How many dimes and how many quarters does Moolah have?

Equations Window Picture Answers & Buttons Pressed

6 y2 =

Day 3- Solving Systems by Substitution

Solving Systems of Equations Using Substitution

Steps: *Note, you can choose which equation will be “equation A” and which will be “equation B” 1) Solve equation A for one of the variables (if needed). 2) Using your new equation A, substitute into equation B. This new equation should have only one variable. 3) Solve for the variable in your new equation from step 2. 4) Substitute your answer from step 3 into the new equation A to find the other variable. 5) Check your answer in both original equations 6) Write a solution set. If your variables are x and y, make an ordered pair.

Solve each system by substitution and check your answers. **Look for two problems with no solution & infinite solutions. 1. a = 3b + 1 5b – 2a = 1

2. -2x – y = -5 y = - ½x – 4

7 3. x + 2y = -4 y + 3x = 3

4. x = -2y + 4 3.5x + 7y = 14

5. y + 11 = 3x y – 3x = -13

8 6. A snack bar sells two sizes of snack packs. A large snack pack is $5, and a small snack pack is $3. In one day, the snack bar sold 60 snack packs for a total of $220. How many small snack packs did the snack bar sell?

9 7. The population of math island is 2,500. If the number of males is 240 more than the number of females, how many males and females are there on math island?

Day 4- Solving System by Elimination

Quiz 1: Days 1 – 2 (Graphing Systems)

Solving Systems by Elimination In the elimination method for solving systems, you use the properties of addition and subtraction of equality to eliminate a variable in the system. The elimination method is the preferred method of solving when a variable in one of the two equations cannot be easily isolated.

Steps for solving simultaneous equations using elimination: 1) Line up like variables in the two equations. 2) Determine which variable you will eliminate. 3) Multiply (if necessary) to eliminate one variable. Combine equations. 4) Solve for one variable. 5) Solve for the other variable by substituting in an original equation. 6) Check in both original equations.

10 Solve each system by elimination and check your answers. **Look for two problems with no solution & infinite solutions.

1) x + y = 2 2) x + 2y = 3 x – y = 4 x – y = 6

Check 1: Check 2: Check 1: Check 2:

3) 2x – 4y = -6 4) 3x – 2y = -3 x – y = -1 6y = 9x + 9

11 5) 5x + y = 15 6) 3x – 2y = 8 3y = -15x + 6 2x – 2y = 5

7) A farm raises a total of 220 chickens and 8) A student bought 3 boxes of pencils and pigs. The number of legs of the stock in the 2 boxes of pens for $6. He then bought farm totals 520. How many chickens and 2 boxes of pencils and 4 boxes of pens pigs are on the farm? for $8. Find the cost of each box of pencils and each box of pens.

12 Day 5- Solving Harder Systems of Equations by Elimination

Double Elimination Problems: Solve each system by elimination and check your answers. 1) 5x – 2y = 8 2) 4m + 3p = -1 3x – 7y = -1 5m + 4p = 1

13 3) 4a + 5b = 35 4) 3x + 7y = -2 3a + 4b = 27 2x + 3y = -3

5) The cost of admission to the Spring 6) Blue Bison soccer team goes out for Carnival was $39 for a group of 8 pizza after a game. Five large pizzas children and 3 adults. The admission plus three medium pizzas together have was $43 for another group of 11 110 slices. Two large pizzas plus nine children and 2 adults. What was the medium pizzas have 122 slices. How admission price for each child? What many slices are in each size? was the admission price for each adult?

14 Day 6: Application Problems involving Systems of Equations

WARM – UP Decide on the most appropriate method for solving:

You and your friend were working the ticket table at the last school basketball game. The tickets sold for $5 for adults and $3 for students. A total of 375 people attended the game and a total of $1375 was taken in at the ticket table. How many adult tickets were sold and how many student tickets were sold?

15 Applications of Systems

Systems of Equations are useful for modeling problems involving mixtures, rates, and break-even points.

The break – even point for a business is the point at which Income equals expenses. The graph shows the break – even point for one such business.

Notice that the values of y on the “expenses” line represent what was spent for the company to run. The values of y on the “income” line represent dollars brought in as income. So y is used to represent both income and expenses.

Finding a Break – Even Point

Find a break-even point by solving for the values that make your expenses equal your income. Base your equations on the two equations below:

Expenses = (expenses per item)(number of items) + (other costs) Income = (gross income per item)(number of items) + (other income) Example 1: A carpenter makes and sells rocking chairs. The materials for each chair cost $22.50. The chairs sell for $75 each. If the carpenter spends $420 on advertising, how many chairs must she sell to break even?

Example 2: A parent teacher organization is selling baskets of cookies for a school fundraiser. The materials needed to make each basket are $3.75, and the baskets are being

16 sold for $10 each. If they spend $75 to advertise their sale, how many baskets must be sold to break even?

Example 3: I am going into business selling tomatoes. It costs $.56 per pound to grow the tomatoes, plus $520 rent each month to rent the ground. If I sell the tomatoes for $1.60 per pound, how many pounds of tomatoes must I sell each month to break even? If I sell 980 pounds, will I make or lose money?

Day 7 – Linear Inequalities

WARM-UP Quiz from Days 3 - 5

Understanding Linear Inequalities  A linear inequality in two variables such as y x – 3, can be formed by replacing the equals sign in y = x – 3 with an inequality symbol.  A solution of the inequality is an ordered pair that makes the inequality true ex. (1, 2).  A linear inequality really has an infinite number of solutions that can be represented in the coordinate plane as the set of all points on one side of a boundary line.

17  All points on one side of the line are a solution while all points on the other side are not. Symbol Symbol Translated into Type of Line Would a Point Shade Above or Used Words Drawn on this line be Below Line a solution? y < dashed No Below y > Y

Graphing an Inequality in Two Variables

1. y < x + 2 a) Is (2, 4) a solution? 2. y > 3x – 4 a) Is (-1, -5) a solution?

b) b) List a point in the List a point in the solution. solution. c) c) List a point that is List a point that is not in the solution. not in the solution.

3. x+ y � 3 4. x- 2 y � 10

18 Name a point in the solution: ______Name a point not in the solution: ______

Graphing a Linear Inequality in One Variable on the Coordinate Plane

5. Graph x > 2 6. Graph y -3

a.) Is (2, -3) a solution? ______a.) Is (2, -3) a solution? ______b.) Is (0, 2) a solution? ______b.) Is (0, 2) a solution? ______c.) Is (15, -6) a solution? ______c.) Is (15, -6) a solution? ______7.

Day 7 Homework – Linear Inequalities

21 Day 8 – Systems of Linear Inequalities

WARM –UP

1. Is the ordered pair (-5, 2) a solution of y < x + 2?

2. Write the inequality for the graph below. 3. You are writing an inequality from a graph. The grey indicates the shaded region. The boundary line is dashed and has a slope of -3 and a y intercept of 2. The area above the line is shaded. What inequality should you write?

Systems of Linear Inequalities A system of linear inequalities is made up of two or more linear inequalities. A solution is an ordered pair that makes each of the linear inequalities in the system true. When graphing a system of inequalities, the solution is the region where the graphs of the individual inequalities overlap, which is usually denoted with an “S” for solution set. Graphing a System of Inequalities Example 1: Graph and check the following system of inequalities: y�+ x 5 and -3x + y � 4

Using the calculator: y�+ x 5 and -3x + y � 4 together makes:

The “double-shaded” region is the solution to the system

22 Example 2: Graph the system of inequalities. & 5x+ 3 y � 9 x � 3 Steps

Check:

Solve each of the systems below by graphing.

Example 3: Example 4: + - > 4 + x – 4y > 0 + < y � 1 x4 y 12 0 & 2x y 4 &

Circle each point below that represents a Circle each point below that represents a solution to the system above: solution to the system above:

23 (-4, 0) (0, 7) (7, 2) (4, 2) (-7, 7) (5, -2) (-2,- 2) (3, -2) * Discussion: 1. When you say that there is no solution to a system of inequalities, what do the lines look like?

2. How is finding a solution to a system of inequalities different from finding a solution to a system of equations? How is it the same?

Determine if a Point is in the System of Inequalities Example 5: Determine whether (1, 4) is a solution to the system: 2x + y > 3 -3x - y 5

24 Day 8 Homework – Systems of Linear Inequalities

Solving Linear Inequality Systems Graphically

x  1 y  x #1. Graph  #2. Graph  y  2 y  2x  3

–8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x –1 –1

–2 –2

–3 –3

–4 –4

–5 –5

–6 –6

–7 –7

–8 –8

2x  y  2 x  y  2  #3. Graph  #4. Graph x  3 x  y  2  y  1 y  y

–8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x –1 –1

–2 –2

–3 –3

–4 –4

–5 –5

–6 –6

–7 –7

–8 –8 25  2 y   x 1  3  x  3y  12 #5. Graph  #6. Graph  3 y  x  4 y  x 1   4

–8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x –1 –1

–2 –2

–3 –3

–4 –4

–5 –5

–6 –6

–7 –7

–8 –8

Write a system of inequalities for the dark region of each graph.

#7. ______#8. ______

______

-5 -4 -3 -2 -1 1 2 3 4 5

-5 Day 9: Systems Application Problems-3 Applications of Linear Inequalities

/Example 1: You are buying paperback and hardcover books at -4 a book sale. Paperbacks cost $2.50 and Hardcover books cost $4.00. You can spend at most $20.

a.) /Set up let statements.

b.) Create an inequality and graph on the grid

provided. Question: Why doesn’t the graph show any

negative x or y values?

c.) Find one possible combination for the number

of paperback and hardcover books you could buy.

d.) Now suppose you must spend at least $5 in paperbacks.

Create an additional inequality and graph on the same

grid. Now name a possible combination based on both

constraints.

Example 2: Marsha is buying plants and soil for her garden. The soil costs $4 per bag, and the plants cost $10 each. She wants to buy at least 5 plants and can spend no more than $100. 27 / a) Write a system of linear inequalities to model the situation.

Let x = number of bags of soil Let y = number of plants

b) Graph the equations to illustrate your answer. Example 3: Set up a system of inequalities for the problem below and graph the two inequalities on the grid provided to help answer the question.

28 /Example 4:

29 Day 9 Homework: Application Problems with Systems

30 /1. Suppose you intend to spend no more than $60 buying books. Hardcover books cost $12 and paperback cost $5. You can spend at most $20 on paperback books. How many books of each type can you buy? Develop the inequalities and graph them on the graph paper provided. y 12 Let x = number of hardcover books 11 Let y = number of paperback books 10 9

1 2 3 4 5 6 7 8 9 10 11 12 x 2. Miss Jones wants to buy at least 5 books. Each paperback book costs an average of $10 and each hardcover book costs an average of $20. Miss Jones is planning to spend less than $140 on books.

a) Write and graph a system of inequalities to model the number of books she can purchase to stay under budget. Be sure to label all parts of your graph. b) State one solution that would work. How much did she spend on hardcover books?

Let x = number of paperback books Let y = number of hardcover books

31 Day 10- Review Day

1. When solved graphically, which system of equations will have exactly one point of intersection?

a. y = -x + 15; y = -x + 25 b. y = .5x + 30; y = .5x – 30

/c. y = 3/5x +12; y = 0.6x – 29 d. y = - x – 20; y = x + 17

2. What is the solution to the system graphed below?

a. (1, 0) and (-3, 0)

b. (0, -3) and (0, -1)

c. (-1, -2)

d. (-2, -1)

3. Tell whether the system has one solution, infinitely many, or no solution. Does it make the system consistent or inconsistent? If consistent, is it dependent or independent? x – 3y = -7 2x = 6y – 14

4. /Determine the solution to the following system of equations graphically: y=2 x - 3 1 y= x + 6 2

32 5. Solve the system of equations algebraically:

6. For the following system, what is the x – value of the solution? -x + 2y = 6 6y = x + 18

1 7. Solve the system: x- y = 9 3 y=3 x - 24

6x+ 3 y = 36 8. Solve the system: -3x + 5 y = 34

33 9. You pay $22 to rent 6 video games. The store charges $4 for a new game and $2 for older games. How many new games did you rent?

10. Set up and solve the following problem as a system of equations. //

34 11. The senior classes at Shaker and Ravena both planned trips to Great Escape. Shaker rented and filled 8 vans and 8 buses with 240 students. Ravena rented and filled 4 vans and 1 bus with 54 students. Every van has the same number of students in it as did the buses. Find the number of students in each van and bus.

12. A concession stand sold a total of 138 small and large popcorns. A small popcorn costs $2.50, and a large costs $4.00. Total popcorn sales were $466.50. How many large popcorns were sold?

35 13. Next weekend Marnie wants to attend either carnival A or carnival B. Carnival A charges $6 for admission and an additional $1.50 per ride. Carnival B charges $2.50 for admission and an additional $2 per ride. a) In function notation, write A(x) to represent the total cost of attending carnival A and going on x rides. In function notation, write B(x) to represent the total cost of attending carnival B and going on x rides.

b) Marnie wants to know how many rides she can go on such that the total cost of attending each carnival is the same. Find the number of rides and the total cost.

c) Marnie wants to go on five rides. Determine which carnival would have the lower total cost. Justify your answer.

14. You go to a garage sale. All the items are either $1 or $5. You spend less than $45. a.) Write and graph a linear inequality that models the situation.

b.) State two possible combination of items you could purchase and still be within your budget?

36 15. Determine whether (0, 1) is a solution to the system of inequalities: 1- x 3 y 3y- 1 > 2 x

16. / a.) Solve the system of inequalities by graphing: y�+ x 4 y y>4 x - 1 8

3 2 b.) Circle each of the ordered pairs that 1 represent solutions to the system above.

–8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 x –1

–2 (-2, 0) (1, 3) (2, 0) (0, 4)

–3 –4 (7, 2) (0, 15) (4, 0) (-1, 5) –5