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Algebra 2 Outcome 1
1
Outcome 1: Linear Relations and Functions
Students will solve two- and three-variable systems of equations algebraically and solve linear
equations and inequalities graphically.
Component 1Create equations in two or more
variables to represent relationships between quantities in a real world
situation. Solve the system of equations for the variable
algebraically and graphically.
Outcome 1 Component 1
Quiz
Component 2Apply linear programming and interpret solutions as viable or
nonviable in a modeling context.
Outcome 1 Component 2
Quiz
Component 3Graph piecewise functions, including
step functions and absolute value functions, translate graphs based on
a parent functions and identify domain, range, intercepts, end behavior, and open and closed
circles on the graphs.
Outcome 1 Component 3
Quiz
Component 4Solve and apply in real-world
context, systems of equations in three variables by hand and using
technology.
Outcome 1 Test Review
Outcome 1 Test
Monday Tuesday Wednesday Thursday FridayAugust 31 September 1 2 3 4
In-Class: In-Class: In-Class: In-Class:
Homework: Homework: Homework: Homework:
7 8 9 10 11
No School – Labor Day
In-Class: In-Class: In-Class: In-Class:
Homework: Homework: Homework: Homework:
14 15 16 17 18In-Class: In-Class: In-Class: In-Class: In-Class:
Homework: Homework: Homework: Homework: Homework:
21 22 23 24 25In-Class: In-Class: In-Class: In-Class: In-Class:
Homework: Homework: Homework: Homework: Homework:
28 29 30 October 1 2In-Class: In-Class: In-Class: In-Class: In-Class:
Homework: Homework: Homework: Homework: Homework:
5 6 7 8 9In-Class: In-Class: In-Class: In-Class:
No School – Teacher InstituteHomework: Homework: Homework: Homework:
12 13 14 15 16
No School – Columbus Day
In-Class: In-Class: In-Class: In-Class:
Homework: Homework: Homework: Homework:
2
O1C1 part 1: Graphing Linear Equations Review Activity
Group 1 Writing an equation given the slope and y−¿ intercept and then graph this equation.
a. Write an equation in slope-intercept
form for the line with a slope of 34 and
a y−¿ intercept of −2. Then graph the equation.
b. Write an equation in slope-intercept form for the line with a slope of −2 and a y−¿ intercept of 3. Then graph the equation.
Equation: ____________________
x
y
Equation: ____________________
x
y
Group 2 Rewrite the equations in slope-intercept form.
a. 3 x+2 y=6 b. 4 x−5 y=15
3
Group 3 Graph Linear Equations
a. y=−3 b. x=5
x
y
.
x
y
Group 4 Write an Equation in slope-intercept Form
What is the equation in slope-intercept form for the line shown?a. b.
x
y
x
y
4
O1C1 part 1: Graphing Linear Equations Review
An equation of the form y=mx+b, where m is the slope and b is the y−¿ intercept, is in slope-intercept form. The variables m and b are called parameters of the equation. Changing either value changes the equation’s graph.
Example 1: Write an equation in slope-intercept form for the line with a slope of 34 and a y−¿
intercept of −2. Then graph the equation, and state the intercepts, domain and range.
y=mx+b Slope-intercept form
y=34
x+(−2) Replace m with 34 and b with −2
y=34
x−2 Simplify
Now graph the equation:Step 1 Plot the y−¿ intercept (0 ,−2)
Step 2 The slope is riserun
=34 . From (0 ,−2), move up 3 units and right 4 units. Plot the
point
Step 3 Draw a line through the two points
5
Intercepts: (0 ,−2) and (83
,0)
If the x−¿ intercept is not exasct, plug in 0 for y in the equation and solve for x
Domain: All real numbers
Range: All real numbers
End Behavior:x→−∞ as y →−∞x→+∞ as y →+∞
Example 2: Graph 3 x+2 y=6. Then state the intercepts, domain and range.
Rewrite the equation in slope-intercept form:3 x+2 y=6 Original equation
3 x+2 y−3x=6−3 x Subtract 3 x from each side
2 y=6−3 x Simplify
2 y=−3 x+6 Rewrite so in the form y=mx+b2 y2
=−3 x2
+ 62 Divide each side by 2
y=−32
x+3 Simplify
Now graph the equation:Step 1 Plot the y−¿ intercept (0,3)
Step 2 The slope is −32 . From (0,3), move down 3 units and right 2 units OR move up 3
units and left 2 units (see below). Plot the point.
Step 3 Draw a line through the two points
Intercepts: (0,3) and (2,0)
Domain: All real numbers
Range: All real numbers
End Behavior:x→−∞ as y →+∞x→+∞ as y →−∞
Study Tip
6
Counting and Direction When counting rise and run, a negative sign may be associated with the value in the numerator or denominator. If placing the negative on the numerator, begin by counting down for the rise. If placing the negative on the denominator, count left when counting the run. The resulting line will be the same.
Except for the graph of y=0, which lies on the x−¿ axis, horizontal lines have a slope of 0. They are graphs of constant functions, which can be written in slope-intercept form as y=0x+b or y=b, where b is any number. Constant functions do not cross the x−¿ axis. Their domain is all real numbers, and their range is b.
Example 3: Graph y=−3. Then state the intercepts, domain and range.
Vertical lines have no slope. So, equations of vertical lines cannot be written in slope-intercept form.
Now graph the equation:Step 1 Plot the y−¿ intercept (0 ,−3)
Step 2 The slope is 0. Draw a vertical line through the y−¿ intercept of −3
Intercepts: (0 ,−3)
Domain: All real numbers
Range: y=−3
End Behavior:x→−∞ as y →−3x→+∞ as y →−3
There are times when you will need to write an equation when given a graph. To do this, locate the y−¿ intercept and use the rise and run to find another point on the graph. Then write the equation in slope-intercept form. You may also pick two points on the graph and use the slope formula to compute
rise over run (think back to algebra 1). In case you forgot, the formula is m=y2− y1
x2−x1
Example 4: Write an Equation in Slope-Intercept Form
What is the equation in slope-intercept form for the line shown?
7
Step 1 The line crosses the y−¿ axis at (0,1), so the y−¿ intercept is 1 and therefore, b=1
Step 2 To get from (0,1) to (3,0), go down 1 unit and 3 units to the right. The slope is −13
Step 3 Write the equation: y=mx+b so y=−13
x+1
HOMEWORK: Graph each equation. Then state the intercepts, domain and range. When an equation is not written in slope-intercept form, it may be easier to rewrite it before graphing. 1. 3 x−4 y=12
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End Behavior:
x
y
8
2. −2 x+5 y=10
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End Behavior:
x
y
3. y=5
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End Behavior:
x
y
4. 2 y=1
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End Behavior:
x
y
9
5. What is an equation in slope-intercept form for the line shown?
10
O1C1 part 2: Graphing Lab – Intersection of Two Graphs
Graph both equations in the standard viewing window:
3 x+ y=9x− y=−1
1. Rewrite each equation in the form y=mx+b 3 x+ y=9 becomes y=−3x+9 x− y=−1 becomes y=x+1
2. Enter −3 x+9 as "Y1” and x+1 as "Y2”. 3. Make sure you have a standard window. Press “ZOOM” then “ZStandard” or “6”.
4. When you do step #3, this should take you to the graph, but if it doesn’t, press “GRAPH”.
5. To find the intersection of the lines, press “2nd" “CALC” “Intersect” or “5” then “ENTER” “ENTER” “ENTER” (3 times). The intersection is (2,3).
Use the graphing calculator to find where each pair of graphs intersect. 1. 2 x+4 y=36
10 y−5 x=0
Answer: _______________
2. 2 y−3 x=7
5 x=4 y−12
Answer: _______________
3. 4 x−2 y=16
7 x+3 y=15
Answer: _______________
4. 5 x+ y=13
3 x=15−3 y
Answer: _______________
5. 4 y−5=20−3 x
4 x−7 y+16=0
Answer: _______________
6.14
x+ y=114
x−12
y=2
Answer: _______________
7. 3 x+2 y=−3
x+13
y=−4
Answer: _______________
11
Graphing Lab – Intersection of Two Inequalities
Graph both equations in the standard viewing window:y ≥−3 x+4y ≤2 x−1
1. All Ti-84’s have an app called “Inequalz” and that allows us to change the equal sign to an inequality when in the “Y=” screen. To make sure it is turned on, press “APPS”, then scroll down to “Inequalz”, press “ENTER”. It brings up the next screen, do as it says and press any key. You are now ready to use the inequality while graphing.
2. Enter −3 x+4 as “Y1”. Since y ≥−3 x+4, we shade above the line. To make the calculator do this, press “Y=”. While in the screen, when the cursor is on the “=” the bottom menu comes up. To change “Y1” to ≥, press “ALPHA” “GRAPH” (the button right below the ≥). Arrow to the right “”, then type in the inequality.
3. Enter 2 x−1 as "Y2”. Since y ≤2 x−1, we shade below the line. Do the same steps as you did to enter y ≥−3 x+4 only you will press “ALPHA” “ZOOM” instead.
4. When both inequalities are entered, press “GRAPH” to see the graph.
Use the graphing calculator to solve each system of inequalities, and then match the system with the correct graph. Choices are on the next page.8. y ≥3
y ≤−x+1
Answer: _______________
9. y ≥−4 xy ≤−5
Answer: _______________
10. y ≥2−xy ≤ x+3
Answer: _______________
11. y ≥2 x+1
y ≤−x−1
Answer: _______________
12. 2 y≥ 3 x−1
3 y ≤−x+7
Answer: _______________
13. y+5 x≥ 12
y+3≤10
Answer: _______________
14. 5 y+3 x≥ 11
3 y−x ≤−8
Answer: _______________
15. 10 y−7 x ≥−19
7 y−5 x ≤11
Answer: _______________
16.16
y−x≥−3
15
y+x≤ 7
Answer: _______________
12
a. b. c.
d. e. f.
g. h. i.
13
14
O1C1 part 2: Graphing Systems of Equations and Inequalities Review
Work with a partner to perform the following steps for solving a linear system using graph and check.1. Graph both equations in the same coordinate plane.2. Estimate the coordinates of the point of intersection.3. Check the coordinates algebraically in each equation.
Example 1: y=x−4
y=13
x−2
Example 2: x− y=8
3 y+2x=6
x
y
x
y
Example 4: x+ y=2
y=4 x+7
x
y
15
Work with a NEW partner to perform the following steps for solving a system of linear Inequalities. 1. Graph both inequalities in the same coordinate plane.2. Estimate the coordinates of the point of intersection.3. Check the coordinates algebraically in each equation.
Example 6: −x+ y>1
x+ y>3
x
y Is (1,5) a viable solution?YES NO
16
Example 7: 2 x+ y<3
x< y+4
x
y Is (−2,4) a viable solution?YES NO
Example 8: 3 y+9 x<3
y ≥2
x
y Is (2 ,−3) a viable solution?YES NO
Example 9: y>−2 x+6
y ≤ 14
x+5
x
y Is (1,3) a viable solution?YES NO
17
Example 10: y<x+3
y>2 x−1
x
y Is (−3 ,−2) a viable solution?YES NO
18
O1C1 part 2: Graphing Systems of Equations and Inequalities Practice
Solve each by graphing each system.1. y=x+2
y=−x−3
2. y=−x+4
y=x+2
3. x+ y=5
y=3 x−2
x
y
x
y
x
y
4. x−2 y<3
2 x+ y>8
List a viable solution _______
5. −3 x+ y<3
x+ y>−1
List a viable solution _______
6. x+2 y>4
2 x− y>6
List a viable solution _______
x
y
x
y
x
y
7. 2 x+ y=2 8. y=3 x−3 9. y=5 x−1
19
x− y=3 y=−2x+2 y=7−3 x
x
y
x
y
x
y
10. y ≥2 x−2
y ≤−3 x
List a viable solution _______
11. x+ y>2
2 x− y<1
List a viable solution _______
12. y>3 x+2
y ≤−2 x+1
List a viable solution _______
x
y
x
y
x
y
O1C1 part 3: Solving Systems of Equations Algebraically Practice
20
Solve each system and check your answer.1. y=3 x
5 x+ y=242. y=2x+5
3 x− y=4
3. x=8+3 y2 x−5 y=8
4. 3 x+2 y=71y=4+2x
5. 4 x−5 y=92x=7 y
6. y=3 x+8x= y
21
7. 8 x+3 y=262 x= y−4
8. x−7 y=133 x−5 y=23
9. 3 x+ y=192 x−5 y=−10
10. 5 x− y=203 x+ y=12
11. x+3 y=7x+2 y=4
12. 3 x−2 y=113 x− y=7
22
13. 7 x+ y=295 x+ y=21
14. 8 x− y=176 x+ y=11
15. 9 x−2 y=506 x−2 y=32
16. 7 y=2 x+353 y=2 x+15
17. 2 y=3 x−12 y=x+21
18. 19=5x+2 y1=3 x−4 y
23
24
O1C1 part 4: Writing Systems of Equations Activity
You will work in partners to define the variables then write the systems of equations for the following 10 problems. DO NOT SOLVE. You will have approximately 15 minutes to write all of your equations. After the allotted time, get together with another pair and compare your equations for each problem. If necessary, make any corrections to your equations.
1. Rob is comparing the costs of long distance calling cards. To use card A, it costs $ 0.50 to connect and then $ 0.05 per minute. To use card B, it costs $ 0.20 to connect and then $ 0.08 per minute.
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
2. The diagrams show the birth lengths and growth rates of two species of shark. If the growth rates stay the same, at what age would a Spiny Dogfish and a Greenland shark be the same length?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
3. Joe and Wanda sell home theater systems. Joe earns a base salary of $2400 per month, plus $ 100 for each system he sells. Wanda earns a base salary of $2200 per month plus $120 for each system she sells.
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
25
4. You and a friend are both reading a book. You read 2 pages each minute and have already read 55 pages. Your friend reads 3 pages each minute and has already read 35 pages. How long will it take for you and your friend to read the same number of page, and how many pages will that be?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
5. A store sells small notebooks for $8 and large notebooks for $ 10. If you buy 6 notebooks and spend $56, how many of each size notebook did you buy?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
6. Roberto is competing in a bicycle race. He has traveled 12 miles and is maintaining a steady speed of 15 mph. Alexandra is competing in the same race but got a flat tire. She has traveled 8 miles and is maintaining a steady speed of 18 mph.
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
7. A shop has 1 pound bags of peanuts for $ 2 and 3 pound bags of peanuts for $5.50. If you buy 5 bags and spend $ 17, how many of each size bag did you buy?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
26
8. Lynn is piloting a plane at an altitude of 10,000 feet. She begins to descend at a rate of 200 feet per minute. At the same time that Lynn begins to descend, Miguel is flying a different plane at an altitude of 5000 feet. At the same time that Lynn begins to descend, Miguel begins to climb at a rate of 50 feet per minute.
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
9. You can choose between two tennis courts at two university campuses to learn how to play tennis. One campus charges $25 per hour. The other campus charges $ 20 per hour plus a one-time registration fee of $ 10. Find the number of hours for which the costs are the same.
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
10. Juan is comparing cell phone plans show in the advertisement.
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
27
28
O1C1 part 4: Writing and Solving Systems of Equations Word Problems Practice
Define the variables, write the system of equations, and then solve. Write your answer as a statement.1. Suppose you bought eight oranges and one grapefruit for a total of $4.60. Later that day, you
bought six oranges and three grapefruits for a total of $4.80. What is the price of each type of fruit?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
Solution: _________________________
_________________________
2. You can buy DVD’s at a local store for $ 15.49 each. You can buy them at an online store for $13.99 each plus $6 for shipping. How many DVD’s can you buy for the same amount at the two stores?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
Solution: _________________________
_________________________
3. Last year, a baseball team paid $20 per bat and $12 per glove, spending a total of $1040. This year, the prices went up to $25 per bat and $ 16 per glove. The team spent $ 1350 to purchase the same amount of equipment as last year. How many bats and gloves did the team purchase each year?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
Solution: _________________________
_________________________
29
4. If the perimeter of the square at the below is 72 units, what are the values of x and y?
Equation 1: _________________________
Equation 2: _________________________
Solution: _________________________
_________________________
5. A student has some $1 bills and $5 bills in this wallet. He has a total of 15 bills that are worth $47. How many of each type of bill does he have?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
Solution: _________________________
_________________________
6. Jenny’s Bakery sells carrot muffins at $2 each. The electricity to run the oven is $ 120 per day and the cost of making one carrot muffin is $ 1.40. How many muffins need to be sold each day to break even?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
Solution: _________________________
_________________________
30
7. In the final round of a singing competition, the audience voted for one of the two finalists, Luke or Sean. Luke received 25 % more votes than Sean received. Altogether, the two finalists received 5175 votes. How many votes did Luke receive?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
Solution: _________________________
_________________________
8. Jim invests money into two separate accounts and earns a total of $240 interest in one year. Account 1 paid 6 % interest and Account 2 paid 3%. If the total amount invested was $5000, how much was in each account?
x: _________________________
y: _________________________
Equation 1: _________________________
Equation 2: _________________________
Solution: _________________________
_________________________
31
32
O1C2: Linear Programming Vocabulary Matching
Match each word in the left hand column with the definition in the right hand column.
1. Linear Programming a. Capable of working successfully
2. Feasible Region b. The highest value in a given set of data
3. Optimize c. The lowest value in a given set of data
4. Bounded d. Conditions given to variables, often expressed as linear inequalities
5. Maximum e. The points of intersection of the graphs of the constraints that determine a feasible region
6. Minimum f. A region when the graph of a system of constraints is a polygonal region
7. Objective Function g. The linear function to be maximized or minimized
8. Constraints h. To seek the best price or amount to minimize costs or maximize profits
9. Vertices
i. A method for finding maximum or minimum values of a function over a given system of inequalities with each inequality representing a constraint
10. Viable j. The intersection of the graphs in a system of constraints
33
34
O1C2: Linear Programming Application Introduction
By the end of this next unit we will be able to DO IT!!Let find out how!
35
Determine how many of each type of device should be made per shift.
Step 1Let x=¿ number of audio players producedLet y=¿ number of phones produced
Step 2Graph the following inequalities (Constraints) (hint – change the scale of your graph)
x≤ 1500 y ≤1700 x+ y≥ 2000 x≥ 600y ≥800
Steps 3 and 4The system is shown in the graph. Note the vertices of the feasible region.
When you graph more than 2 inequalities, your shaded (feasible/viable) region results in a geometric shape.
List the coordinates of each vertex of the shaded region.
_______________ _______________ _______________ _______________ _______________
Step 5The function to be minimized is:
f (x , y )=55 x+95 y
We call this the OBJECTIVE Function because it is the Objective of the problem.
Step 6Take each of these coordinates and plug them in for x and p into the objective function to find the minimum value because our objective is to KEEP COSTS DOWN!
Step 7Write your answer as a complete sentence.
Is it possible (viable) to produce 800 audio players and 1000 phones? YES or NO
Why or why not?
36
x
y
O1C2: Introduction to Graphing Feasible Region Notes
Situations often occur in business in which a company hopes to either maximize or minimize costs, and may constraints need to be considered. These issues can often be addressed by the use of systems of inequalities in linear programming.
Linear programming is a method for finding maximum or minimum values of a function over a given system of inequalities with each inequality representing a constraint. After the system is graphed and the vertices of the solution set, called the feasible region, are substituted into the function, you can determine the maximum or minimum value.
The feasible region is enclosed, or bounded, by the constraints. The maximum or minimum value of the related function always occurs at a vertex of the feasible region.
Example 1: Let’s find our first feasible region:
Step 1 Graph the system of inequalities:y ≥3y ≤6
y ≤3 x+12y ≤−2 x+6
x
y
Step 2 Each point of intersection is called a vertex. List the coordinates of each vertex in the chart below to determine where the maximum or minimum occurs.
Step 3 Take each of these coordinates and plug them in for x and y into the objective function to find the maximum or minimum value.
The objective function is: f ( x , y )=4 x−2 yVertices 4 x−2 y f (x , y ) Max or Min
37
38
O1C2: Introduction to Graphing Feasible Region Practice
Find the values of x and y that maximizes or minimizes the objective function for each graph. Then find the maximum or minimum value. List the vertices and plug them into the objective function to determine which value is the max or min.
1. Maximum for P=2 x+3 y 2. Minimum for C=x+2 y 3. Maximum for P=3 x+ y
Graph each system of constraints. Name all vertices. Then find the values of x and y that maximizes or minimizes the objective function.
4. {x+2 y ≤6x ≥2y≥ 1
Minimum for C=3 x+4 y
5. { x+ y ≤ 5x+2 y ≤8
x≥ 0y ≥ 0
Maximum for P=x+3 y
6. { x+ y ≤62x+ y ≤10
x≥ 0y ≥ 0
Maximum for P=4 x+ y
x
y
x
y
x
y
39
7. {3x+2 y ≤62x+3 y ≤6
x≥ 0y ≥ 0
Maximum for P=4 x+ y
8. { x+ y ≤54 x+ y ≤ 8
x ≥ 0y ≥0
Minimum for C=x+3 y
x
y
x
y
40
O1C2: Linear Programming – Writing Constraints
1. Wood pulp can be converted to either notebook paper or newsprint. The Canyon Pulp and Paper Mill can produce at most 200 units of paper a day. Regular customers require at least 10 units of notebook paper and 80 sheets of newspaper daily. If the profit on a unit of notebook paper is $ 500 and the profit on a unit of newsprint is $ 350, how many units of each type of paper should the mill produce each day to maximize profits?
a. What is the final question looking for?
b. So x=¿ _________________________ y=¿ _________________________
c. What are the profits of x and y?
x=¿ _________________________ y=¿ _________________________
d. Use these to write the OBJECTIVE FUNCTION: P=¿ _________________________
e. What are the constraints? Look for “at most” or “at least” and then write the functions
2. A carpenter makes book cases in two sizes, large and small. It takes 6 hours to make a large bookcase and 2 hours to make a small one. The profit on a large bookcase is $ 50 and the profit on a small one is $20. The carpenter can spend only 24 hours per week making bookcases and must make at least two of each size per week. The carpenter wants to know how many of each size must be made per week in order to maximize profit.
a. What is the final question looking for?
b. So x=¿ _________________________ y=¿ _________________________
c. What are the costs of x and y?
x=¿ _________________________ y=¿ _________________________
d. Use these to write the OBJECTIVE FUNCTION: P=¿ _________________________
e. What are the constraints? Look for “at most” or “at least” and then write the functions
41
3. The Cookie Factory’s bestselling items are chocolate chip cookies and peanut butter cookies. They want to sell both types of cookies together in combination packages. The different sized packages will contain between 6 and 12 cookies, inclusively. At least three of each type of cookie should be in each package. The cost of making a chocolate chip cookie is $0.19, and the selling price is $0.44. The cost of making a peanut butter cookie is $0.13, and the selling price is $0.39. How many of each type of cookie should be in each package to maximize profit?
a. x=¿ _________________________ y=¿ _________________________
** Determine the profit by performing a calculation (se lling price−making price)
b. Use these to write the OBJECTIVE FUNCTION: P=¿ _________________________
c. What are the constraints? Look for “between” or “at least” and then write the functions
4. A tourist agency can sell up to 1200 travel packages for a bowl game. The package includes air fare, weekend accommodations, and the choice of two types of flights: nonstop flight or flight with two stops. The nonstop flight can carry up to 150 passengers, and the flight with stops can carry up to 100 passengers. The agency can locate no more than 10 planes. All packages for full nonstop flight earn $ 180,000, and all packages for full flight with stops earn $135,000. How many of each flight should be scheduled to bring in the most money?
a. x=¿ _________________________ y=¿ _________________________
b. Use these to write the OBJECTIVE FUNCTION: P=¿ _________________________
c. What are the constraints? Look for “up to” or “no more” and then write the functions
42
5. A tire manufacturer has 1000 units of raw rubber to use in producing radial tires for passenger cars and tractor tires. Each radial tire requires 5 units of rubber; each tire for tractors requires 20 units. Labor costs are $ 8 for a radial tire and $12 for a tractor tire. The manufacturer does not want to pay more than $1500 in labor costs and wants to make a profit of $10 per radial tire and $25 per tractor tire. The manager needs to know how many of each kind of tire to make in order to maximize profits.
x
y
6. The area of a parking lot is 600 square meters. A car requires 6 square meters and a bus requires 30 square meters of space. The lot can handle a maximum of 60 vehicles. If a car cost $3 and a bus cost $ 8 to park in the lot, determine the number of vehicles to maximize the amount collected.
x
y
43
7. A painter has exactly 32 units of yellow dye and 54 units of green dye. He plans to mix as many gallons as possible of color A and color B. Each gallon of color A requires 4 units of yellow dye and 1 unit of green dye. Each gallon of color B requires 1 unit of yellow dye and 6 units of green dye. Find the maximum number of each gallon possible.
x
y
8. Trenton, MI, a small Midwestern town, is trying to establish a public transportation system using 10 or fewer vans and buses. It can spend no more than $100,000 for both types of vehicles and no more than $500 per month for maintenance. Trenton can purchase a van for $ 10,000 and maintain it for $100 per month. The buses cost $ 20,000 each and can be maintained for $75 per month. Each van can carry a maximum of 15 passengers and each bus a maximum of 25 passengers. How many of each vehicle should be used in order to have the most riders?
x
y
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9. Garth’s farm consists of 240 acres of cropland. He wishes to plant this acreage in corn or oats. The total number of hours of labor during the production period is 320. Each acre of land in corn production uses 2 hours of labor while production of oats requires 1 hour per acre. The profit per acre of corn production is $40, and the profit per acre of oats production is $30. How many of each crop should be planted to yield the most profit?
x
y
10. A farmer has 90 acres available for planting millet and alfalfa. Seed costs $4 per acre for millet and $6 per acre for alfalfa. Labor costs will amount to $20 per acre for millet and $ 10 per acre for alfalfa. The expected income from millet is $100 per acre and from alfalfa, $150 per acre. The farmer intends to spend no more than $480 for seed and $1400 for labor. What will be the maximum income?
x
y
45
46
O1C2: Linear Programming Putting it all Together
1. Each week, Mackenzie can make 10 to 25 necklaces and 15 to 40 pairs of earrings. If she earns profits of $3 on each pair of earrings and $5 on each necklace, and she plans to sell at least 30 pieces of jewelry, how can she maximize profit?
Step 1: Variables
Step 2: Constraints (System of Inequalities)
Steps 3 and 4: Graph the System and Identify the Vertices. List the coordinates of each vertex of the shaded region.
Step 5: Objective Function
Step 6: Substitute Vertices into Objective Function
Step 7: Answer
x
y
Is it viable to produce 23 necklaces and 38 earrings? YES or NOWhy or why not?
2. Sarah’s Bakery makes two types of birthday cakes: yellow cake, which sells for $25, and strawberry cake, which sells for $ 35. Both cakes are the same size, but the decorating and assembly time
47
required for the yellow cake is 2 hours, while the time is 3 hours for the strawberry cake. There are 450 hours of labor available for production. How many of each type of cake should be made to maximize revenue?
Step 1: Variables
Step 2: Constraints (System of Inequalities)
Steps 3 and 4: Graph the System and Identify the Vertices. List the coordinates of each vertex of the shaded region.
Step 5: Objective Function
Step 6: Substitute Vertices into Objective Function
Step 7: Answer
x
y
Is it viable to produce 3 yellow cakes and 5 strawberry cakes? YES or NOWhy or why not?
O1C2: Linear Programming Homework
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1. Mrs. Smith has a bakery that is open for 6 hours each day. Her two famous items are apple pies and cherry cobblers. It takes her approximately 2/3 (40 minutes) of an hour to make apple pie and 1 hour to make cherry cobbler. She charges $6 for a pie and $9 for a cobbler. Mrs. Smith cannot do more than 2 cherry cobblers per day. Find the number of each type of pastry that maximizes Mrs. Smith’s income for the day.
Let x=¿ apple pie; y=¿ cherry cobbler
OBJECTIVE FUNCTION: P=6x+9 y
Constraints:23
x+ y≤ 6
y ≤2x≥ 0y ≥0
Graph the system of constraints. Name all vertices. Then find the values of x and y that maximizes Mrs. Smith’s income for the day.
Vertices and work: 2 4 6 8 10
2
4
6
8
10
x
y
Number of each type of pastry: ______________________________________________________
Maximum income: __________________________________________________________________
Is it viable to produce 3 apple pies and 1 cherry cobbler? YES or NO
Why or why not?
2. Your art club wants to sell greeting cards using members’ drawings. Small blank cards cost $ 10 per box of 18. Large blank cards cost $ 15 per box of 12. You make a profit of $ 52.50 per box of small
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cards and $ 85 per box of large cards. The club can buy no more than 264 total cards and spend no more than $210. How can the art club maximize its profit?
x=¿ _________________________
y=¿ _________________________
OBJECTIVE FUNCTION:
P=¿ _________________________
Constraints:
_________________________
_________________________
_________________________
_________________________
Graph the system of constraints. Name all vertices. Then find the values of x and y that determine the maximum profit.
Vertices and work:
2 4 6 8 10 12 14 16 18 20 22
2
4
6
8
10
12
14
16
18
20
22
x
y
Number of each type of card: __________________________________________________________
Maximum profit: ____________________________________________________________________
Is it viable to sell 2 small cards and 10 large cards? YES or NO
Why or why not?
O1C3 part 1: Piecewise Functions Notes
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Piecewise-Defined FunctionsThe function relating income and tax is not a linear function because each interval, or piece, of the function is defined by a different expression. A function that is written using two or more expressions is called a piecewise-defined function.
On the graph of a piecewise-defined function: a dot indicates that the point is included in the graph you will use ≤ or ≥ when writing the
function A circle indicates that the point is not included in the graph you will use ¿ or ¿ when writing the
function
**Remember: f ( x )=¿ means the same thing as y=¿ when plugging in and finding coordinates
Example 1: Graph f ( x )= {x+4 , x>−2 Then state the intercepts, domain, range and end behavior.
Step 1 Graph f ( x )=x+4 for x>−2f ( x )=x+4 so f ( x )= (−2 )+4 therefore, f ( x )=2
Because −2 does not satisfy the inequality, begin with an open circle at (−2,2),
then plot the slope of 11
Step 3
The intercepts are (0,4) The function is defined for all values of x, so the domain is x>−2. The y−¿ coordinates of points on the graph are all real numbers greater than 2, so
the range is y>2 The end behavior is x→+∞ as y →+∞
Example 2: Graph f ( x )={ x−2 ,∧x←1x+3 ,∧x ≥−1 Then state the intercepts, domain, range and end behavior.
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Step 1 Graph f ( x )=x−2 for x←1f ( x )=x−2 so f ( x )=(−1)−2 therefore, f ( x )=−3
Because −1 does not satisfy the inequality, begin with a circle at (−1 ,−3), then
plot the slope of −11 (negative because x←1 so we only want the left hand part
of the graph)Step 2 Graph f ( x )=x+3 for x≥−1
f ( x )=x+3 so f ( x )=(−1)+3 therefore, f ( x )=2
Because −1 does satisfy the inequality, begin with a dot at (−1,2), then plot the
slope of 11 (positive because x≥−1 so we only want the right hand part of the
graph)Step 3
The intercepts are (0,3) The function is defined for all values of x, so the domain is all real numbers. The y−¿ coordinates of points on the graph are all real numbers less than −3 and all
real numbers greater than or equal to 2, so the range is y←3 or y≥ 2 The end behavior is x→−∞ as y →−∞ and x→+∞ as y →+∞
Example 3: Write the piecewise-defined function shown in the graph below:
Examine and write a function for each portion of the graph: The left portion of the graph is the graph of y=2x+3. There is a circle at (1,5), so the
linear function is defined for x<1. The right portion of the graph is the graph of y=−x+2. There are dots at (1,1) and
(2,0), so the linear function is defined for 1≤ x ≤2.
So the piecewise-defined function of the graph above is:
f ( x )={ 2 x+3 ,∧x<1−x+2 ,1≤ x≤ 2
O1C3 part 1: Piecewise Functions Practice
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Graph each function. State the intercepts, domain, range and end behavior.
1. f ( x )={x ,−4< x<2−x+6 , x≥ 2
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
2. f ( x )={ 8 ,∧x≤−1−4−x ,∧x>−1
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
53
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Write the piecewise-defined function shown in each graph. State the intercepts, domain, range and end behavior.3.
f ( x )=¿
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
4.
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f ( x )=¿
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
O1C3 part 1: Piecewise Functions Homework
Graph each function. State the intercepts, domain, range and end behavior.1.
f ( x )={−x ,∧x←25 ,∧x>2
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
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2.
f ( x )={x ,∧−2 ≤x<14 ,∧x≥ 1
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
3.
f ( x )={−2 x ,∧x≤−2x−5 ,∧x>6
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
4.
f ( x )={−x+4 ,−5<¿x←212
x+1 ,∧x≥−2
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
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5.
f ( x )={ x+4 ,∧x←2−3 ,∧−2≤ x≤ 3
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
6.
f ( x )={x+3 ,∧x←2−3 x ,∧x>2
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
7.
f ( x )={ x ,∧x<13 ,1≤∧x≤ 3
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
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8.
f ( x )={−2 x ,∧x≤−1x ,∧x≥ 3
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
x
y
Write the piecewise-defined function shown in each graph. State the intercepts, domain, range and end behavior.9.
y=¿
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
10.y=¿
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
11.
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f ( x )=¿
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
12.f ( x )=¿
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
13.f ( x )=¿
Intercepts: ____________________
Domain: ____________________
Range: ____________________
End behavior: ____________________
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O1C3 part 2: Absolute Value Functions Discovery Activity
1. With your graphing calculator graph the following equations. y=x y=|x|
x
y
x
y
2. What is the difference in the range values for each graph?
How does this relate to the meaning of absolute value?
What are the coordinates of the vertex of the absolute value parent function?
3. y=|x| is called the parent function for absolute value. We will now examine how making changes to the parent function affect the graph.a. Graph y=|x−2| and y=|x+2| on your calculator. Describe in your own words how these values
affects your graph.
What role did the absolute value sign play in this affect?
b. Graph y=|x|+2 and y=|x|−2 on your calculator. Describe in your own words how this value affects your graph.
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c. Graph y=3|x| and y=13|x| on your calculator. Describe in your own words how this value
affects your graph.
How does the leading coefficient relate to what you know about slope?
d. Graph y=−3|x| and y=|−3 x| on your calculator. Describe in your own words how this value affects your graph.
e. Graph y=|x−4|+2 on your calculator. Explain the relationship between the vertex of the graph and the values in the function.
f. Without graphing on your calculator and using what you learning in part e, what is the vertex of the following function: y=2|x+4|−3
Now, using what you know about the leading coefficient and your vertex, graph the function below without using a graphing calculator.
x
y
g. Now graph y=|2x+4|−3 on your calculator. Was it the same as your graph above? If not, what was different and how does it relate to your function that you entered?
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O1C3 part 2: Absolute Value Functions Notes
Another piecewise-linear function is the absolute value function. An absolute value function is a function that contains an algebraic expression within absolute value symbols.
A Bit of Math History – Karl Weierstrass (1815–1897)
At the wishes of his father, Weierstrass studied law, economics, and finance at the University of Bonn, but then dropped out to study his true interest, mathematics, at the University of Münster. In an 1841 essay, Weierstrass first used || to denote absolute value.
Example 1: Graph f ( x )=|2x|−4. State the intercepts, domain, range and end behavior.
Step 1: Create a table of values Step 2: Graph the points and connect them
The intercepts are (−2,0), (0 ,−4), and (2,0) The domain is the set of all real numbers The range is f (x)≥−4 The end behavior is f ( x ) →+∞ as x→−∞ and f ( x ) →+∞ as x→+∞
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Graph each function. State the intercepts, domain, range and end behavior.1. g ( x )=|−3 x|
x
yIntercepts:
Domain:
Range:
End behavior:
2. f ( x )=2|x|
x
yIntercepts:
Domain:
Range:
End behavior:
3. h ( x )=|x+4|
x
yIntercepts:
Domain:
Range:
End behavior:
4. s ( x )=|−2 x|+6
x
yIntercepts:
Domain:
Range:
End behavior:
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O1C3 part 2: Absolute Value Functions Homework
Graph each function. State the intercepts, domain, range and end behavior.1. g ( x )=|x+2|−5
x
y
Intercepts:
Domain:
Range:
End behavior:
2. g ( x )=|x−3|+4
x
y
Intercepts:
Domain:
Range:
End behavior:
3. g ( x )=|x|+6
x
y
Intercepts:
Domain:
Range:
End behavior:
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4. g ( x )=2|x|
x
y
Intercepts:
Domain:
Range:
End behavior:
5. g ( x )=14|x|
x
y
Intercepts:
Domain:
Range:
End behavior:
6. g ( x )=−|x+2|−1
x
y
Intercepts:
Domain:
Range:
End behavior:
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7. g ( x )=2|x−1|−5
x
y
Intercepts:
Domain:
Range:
End behavior:
8. g ( x )=|3 x+9|+2
x
y
Intercepts:
Domain:
Range:
End behavior:
9. g ( x )=12|x+2|+2
x
y
Intercepts:
Domain:
Range:
End behavior:
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69
O1C3 part 3: Step Functions Introduction Activity
With your group, use the photo of the parking rates in Chicago to draw a graph representing the rates.
2 4 6 8 10 12 14 16 18 20 22 24
24681012141618202224262830
x
y
70
time
Cost
($)
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O1C3 part 3: Step Functions Notes
Step Functions and Absolute Value FunctionsUnlike a piecewise-defined function, a piecewise-linear function contains a single expression. A common piecewise-linear function is the step function. The graph of a step function consists of line segments.
The greatest integer function, written f (x)=⟦ x ⟧, is one kind of step function. The symbol ⟦ x ⟧ means the greatest integer less than or equal to x. For example, ⟦3.25 ⟧=3 and ⟦−4.6 ⟧=−5.
Study TipGreatest Integer Function Notice that the domain of this step function is all real numbers and the
range is all integers
Real-World Example Using a Step FunctionExample 1: An automotive repair center charges $ 50 for any part of the first hour of labor, and $35
for any part of each additional hour. Draw a graph that represents this situation.
Understand The total labor charge is $ 50 for the first hour plus $ 35 for each additional fraction of an hour, so the graph will be a step function.
Plan If the time spent on labor is greater than 0 hours, but less than or equal to 1 hour, then the labor charge is $50. If the time is greater than 1 hour but less than 2 hours, then the labor charge is $85 ¿), and so on.
Solve Use the pattern of times and costs to make a table, where x is the number of hours of labor and T (x) is the total labor charge. Then graph.
Check Since the repair center rounds any fraction of an hour up to the next whole number, each segment of the graph has a circle at the left endpoint and a dot at the right endpoint.
Answer the following questions about step functions.
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1. What is the difference between step functions and piecewise functions?
2. How do I know if the endpoints are open or closed?
Graph each function. Identify the domain and range.3. g ( x )=−2 ⟦ x ⟧
x
y
4. h ( x )=⟦ x−5 ⟧
x
y
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5. Springfield High School’s theater can hold 250 students. The drama club is performing a play in the theater. Draw a graph of a step function that shows the relationship between the number of tickets sold x and the minimum number of plays y that the drama club must perform.
x
y
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O1C3 part 3: Step Functions Homework
Graph each function. State the intercepts, domain, range and end behavior.1. g ( x )=⟦ x ⟧−6
x
y
Intercepts:
Domain:
Range:
End behavior:
2. g ( x )=⟦ x+2 ⟧+2
x
y
Intercepts:
Domain:
Range:
End behavior:
3. The charge for renting a bicycle from a rental shop for different amounts of time is shown at the right.a. Identify the type of function that models this situation.b. Write and graph a function for the situation.
Time Price12 hour $6.00
1 hour $10.002 hours $16.00
Daily $ 24.00
x
y
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77
O1C4: Graphing Lab – Matrices
A matrix is a rectangular array of variables or constants in rows and columns, usually enclosed in brackets. In a matrix, the numbers or data are organized so that each position in the matrix has a purpose. Each value in the matrix is called an element. A matrix is usually named using an uppercase letter.
The element −1 is in row 2, column 1, depicted by a21
A=[ 8−17
−23
−8
5−31
664 ] The element −8 is in row 3,
column 2, depicted by a32
A matrix can be described by its dimensions. A matrix with m rows and n columns is an m× n matrix (read “m by n”). Matrix A above is a 3× 4 matrix because it has 3 rows (horizontal) and 4 columns (vertical). a12 refers to an element of A, whereas b12 refers to an element of B.
A graphing calculator can be used to perform operations with matrices. Using a TI83/84 Plus graphing calculator, you can solve a system of linear equations using the
MATRIX function. You can use a graphing calculator to find the inverse matrix so that the solution of the system of equations can be easily determined.
Example 1: Write an inverse matrix for the following system of equations. Then solve the system using a graphing calculator.
2 x+ y+z=13 x+2 y+3 z=124 x+ y+2 z=−1
1. Write a coefficient matrix [A] and an answer matrix [B] for the system.
A=[ 2 1 13 2 34 1 2] and B=[ 1
12−1]
2. Enter matrix A into the calculator. Press “2nd” “x−1” “” “”“ENTER” “3” “ENTER” “3” “ENTER” “2” “ENTER” “1” “ENTER” “1” “ENTER” “3” “ENTER” “2” “ENTER” “3” “ENTER” “4” “ENTER” “1” “ENTER” “2” “2nd” “MODE”
3. You will enter matrix B using the same steps as matrix A
4. To find the answer to this system, we will use the inverse of [A] times [B]. Press “2nd” “x−1” “ENTER” “x−1” “2nd” “x−1” “” “ENTER” “ENTER”, this gives us our answer to the system.
5. The first row represents x=−4, the second row represents y=3, and the third row represents z=6. The solution is (−4,3,6).
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Write a matrix for each system of equations. Then solve with a graphing calculator.1. 3 x+2 y=−4
4 x+7 y=132. 2 x+6=6
6 x−2 y=03. 2 x+2 y=−4
7 x+3 y=10
4. 4 x+6 y=08 x−2 y=7
5. 6 x−4 y+2 z=−42 x−2 y+6 z=102 x+2 y+2 z=−2
6. 5 x−5 y+5 z=105 x−5 z=55 y+10 z=0
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O1C4: Solving Systems of Equations in 3 Variables Using Matrices Practice
Write a matrix for each system of equations. Then solve with a graphing calculator.1. −3a−4 b+2 c=28
a+3b−4c=−312a+3 c=11
2. 3 y−5 z=−234 x+2 y+3 z=7 −2 x− y−z=−3
3. 3 x+6 y−2 z=−62 x+ y+4 z=19−5 x−2 y+8 z=62
4. −4 r−s+3 t=−93 r+2 s−t=3r+3 s−5t=29
5. 3 x+5 y−z=12−2 x−3 y+5 z=144 x+7 y+3 z=38
6. 2a−3b+5c=58−5a+b−4c=−51−6a−8b+c=22
7. −5 x+ y−4 z=602 x+4 y+3 z=−126 x−3 y−2 z=−52
8. 4 a+5b−6c=2−3 a−2b+7c=−15−a+4b+2c=−13
9. −2 x+5 y+3 z=−25−4 x−3 y−8 z=−396 x+8 y−5 z=14
80
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O1C4: Solving Systems of Equations in 3 Variables Notes and Homework
Solving a System of Three Equations by ELIMINATION This will be the sample equation used throughout the instructions:
Equation 1)Equation 2)Equation 3)
3 x+2 y+4 z=112 x− y+3 z=45 x−3 y+5 z=−1
1. Eliminate one of the variables in two of the original equations. 3 x+2 y+4 z=114 x−2 y+6 z=8
7 x+10 z=19
Equation 1Equation 2 times 2New Equation 1
2. Add −3 times the second equation to the third. 5 x−3 y+5 z=−1
−6 x+3 y−9 z=−12−x−4 z=−13
Equation 3Equation 2 times −3New Equation 2
3. Solve the new system of linear equations in two variables. 7 x+10 z=19
−7 x−28 z=−91−18 z=−72
z=4
New Equation 1New Equation 2 times 7
Solve for z
4. Solve for the other variable in this system, in this case x.
x=−3Substitute 4 for z into New Equation 1 or 2 to find x. In either case, you should get the same answer.
5. Substitute x=−3 and z=4 into an original equation and solve for y. 2 x− y+3 z=4
2 (−3 )− y+3 ( 4 )=4y=2
Equation 2Substitute −3 for x and 4 for zSolve for y
Now you know x=−3, y=2, and z=4. You can also write it as (−3,2,4). To check your answer, substitute the values for x back in to each of the original equations.
82
Another way is can be solved is with a graphic organizer like the one below.
83
Solution
Substitute
Equation 4 and 5
Equation 2 and 3
New Equation (5)
Equation 1 and 2
New Equation (4)
Using what you learned in the stations activity example, solve within your group the two problems below for x, y, and z.
1. −5 x+ y−4 z=602 x+4 y+3 z=−126 x−3 y−2 z=−52
2. 4 x+6 y−z=−183 x+2 y−4 z=−24−5 x+3 y+2 z=15
84
3. x+2 y+5 z=−12 x− y+z=23 x+4 y−4 z=14
4. 5 x−4 y+4 z=18−x+3 y−2 z=04 x−2 y+7 z=3
85
Solve using your calculator.5. 2 x+ y−1=−3 z
5 x+ y=8+2 zx=5+ y+9 z
6. x+z=2− y2 x+ y=5x+3 y=14+3 z
86
87
O1C4: Solving Systems of Equations in 3 Variables Activity
Seats closest to an amphitheater stage cost $30. The seats in the next section cost $25, and lawn seats are $20. There are twice as many seats in section B as in section A. When all 19,200 seats are sold, the amphitheater makes $ 456,000. How many seats are there in each section?
Write an algebraic expression for the sentences below.A. Seats closest to an amphitheater stage cost $30. The seats in the next section cost $25, and lawn
seats are $20. If all seats are sold, the amphitheater makes $ 456,000.
B. There are twice as many seats in section B as in section A.
C. There are a total of 19,200 seats in the amphitheater.
Try to combine the expressions together or ELIMINATE one of the variables so that they have a new algebraic expression with only 2 variables. Next, take the new expressions and solve using prior knowledge for the remaining two variables to then come up with the final solution!
88
89
O1C4: Application Problems Practice
1. A friend e-mails you the results of a recent high school swim meet. The e-mail states that 24 individuals placed, earning a combined total of 53 points. First place earned 3 points, second place earned 2 points, and third place earned 1 point. There were as many first-place finishers as second- and third-place finishers combined.a. Write a system of three equations that represents how many people finished in each place.
b. How many swimmers finished in first place, in second place, and in third place?
c. Suppose the e-mail had said that the athletes scored a combined total of 47 points. How many swimmers finished in first place, in second place, and in third place?
d. Explain why this statement is false and the solution is unreasonable.
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2. Nick goes to the amusement park to ride roller coasters, bumper cars, and water slides. The wait for the roller coasters is 1 hour, the wait for the bumper cars is 20 minutes long, and the wait for the water slides is only 15 minutes long. Nick rode 10 total rides during his visit. Because he enjoys roller coasters the most, the number of times he rode the roller coasters was the sum of the times he rode the other two rides. If Nick waited in line for a total of 6 hours and 20 minutes, how many of each ride did he go on?
3. Randy usually gets one of the routine maintenance options at Annie’s Garage. Today however, he needs a different combination of work than what is listed.a. Assume that the price of an option is the same price as
purchasing each item separately. Find the prices for an oil change, a radiator flush, and a brake pad replacement.
b. If Randy wants his brake pads replaced and his radiator flushed, how much should he plan to spend?
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4. Kate invested $ 100,000 in three different accounts and together they earned $6300 in interest. If she invested $30,000 more in account A than account C, how much did she invest in each account?
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