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ALGEBRA 1
UNIT 3
WORKBOOK
CHAPTER 6
FALL 2014
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Algebra 1
Section 6.1 Notes: Graphing Systems of Equations
System of Equations: a set of two or more equations with the same variables, graphed in the same coordinate plane
The ordered pair that is a solution of both equations is the solution of the system. A system of two linear equations can have one
solution, an infinite number of solutions, or no solution.
Consistent: a system of equations that has at least one solution
Independent: a consistent system of equations that has exactly one solution
Dependent: a consistent system of equations that has an infinite number of solutions; this means that there are an unlimited solutions
that satisfy both equations
Inconsistent: a system of equations that has no solution; the graphs are parallel
Example 1: Use the graph to determine whether the system is consistent or inconsistent and if it is independent or dependent.
a)
y = –x + 1
y = –x + 4
b)
y=x–3
y = –x + 1
Solve by Graphing: One method of solving a system of equations is to graph the equations on the same coordinate plane and find their
point of intersection. This point is the solution of the system.
Example 2: Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many
solutions. If the system has one solution, name it.
a)
y = 2x + 3
8x – 4y = –12
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b)
x – 2y = 4
x – 2y = –2
c) x – y = 2
3y + 2x = 9
We can use what we know about systems of equations to solve many real-world problems involving constraints that are modeled by
two or more different functions.
Example 3: Naresh and Diego are having a bicycling competition. Naresh is able to ride 20 miles at the start of the competition and
plans to ride 35 more miles than the previous week each upcoming week. Diego is able to ride 50 miles at the start of the competition
and plans to ride 25 more miles than the previous week each upcoming week. Predict the week in which Naresh and Diego will have
ridden the same number of miles.
1) Write a system of equations to represent the system
3)
2) Graph the system to determine the solution.
Use substitution to check your answer.
Example 4: Alex and Amber are both saving money for a summer vacation. Alex has already saved $100 and plans to save $25 per
week until the trip. Amber has $75 and plans to save $30 per week. In how many weeks will Alex and Amber have the same amount
of money?
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Algebra 1
Section 6.1 Worksheet
Use the graph at the right to determine whether
each system is consistent or inconsistent and
if it is independent or dependent.
1. x + y = 3
x + y = –3
2. 2x – y = –3
4x – 2y = –6
3. x + 3y = 3
x + y = –3
4. x + 3y = 3
2x – y = –3
Graph each system and determine the number of solutions that it has. If it has one solution, name it.
5. 3x – y = –2
3x – y = 0
6. y = 2x – 3
4x = 2y + 6
7. x + 2y = 3
3x – y = –5
8. BUSINESS 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.
a. Graph the system of equations y = 0.5x + 20 and
y = 1.5x to represent the situation.
b. How many treats does Nick need to sell per week to break even?
9. SALES A used book store also started selling used
CDs and videos. In the first week, the store sold 40
used CDs and videos, at $4.00 per CD and $6.00 per
video. The sales for both CDs and videos totaled $180.00
a. Write a system of equations to represent the situation.
b. Graph the system of equations.
c. How many CDs and videos did the store sell in the first week?
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Algebra 1
Section 6.2 Notes: Substitution
In the previous lesson, we learned how to solve a system of equations by graphing. Another method for solving a system of equation
is called substitution.
Example 1: Use substitution to solve the system of equations.
a) y = –4x + 12
b) y = 4x – 6
2x + y = 2
5x + 3y = -1
If a variable is not isolated in one of the equations in the system, solve an equation for a variable first. Then you can use substitution
so solve the system.
Example 2: Use substitution to solve the system of equations.
a) x – 2y = –3
b) 3x – y = –12
3x + 5y = 24
–4x + 2y = 20
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Generally, if you solve a system of equations and the result is a false statement such as 3 = -2, there is no solution. If the result is an
identity, such as 3 = 3, then there are an infinite number of solutions.
Example 3: Use substitution to solve the system of equations.
a) 2x + 2y = 8
b) 3x – 2y = 3
x + y = –2
–6x + 4y = –6
Example 4:
a) A nature center charges $35.25 for a yearly membership and $6.25 for a single admission. Last week it sold a combined
total of 50 yearly memberships and single admissions for $660.50. How many memberships and how many single
admissions were sold?
b)
As of 2009, the New York Yankees and the Cincinnati Reds together had won a total of 32 World Series. The Yankees
had won 5.4 times as many as the Reds. How many World Series had each team won?
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Algebra 1
6.2 Worksheet
Use substitution to solve each system of equations.
1. y = 6x
2x + 3y = –20
4. y = 2x – 2
y=x+2
7. x + 2y = 13
–2x – 3y = –18
10. 2x – 3y = –24
x + 6y = 18
13. 0.5x + 4y = –1
x + 2.5y = 3.5
2. x = 3y
3x – 5y = 12
5. y = 2x + 6
2x – y = 2
8. x – 2y = 3
4x – 8y = 12
11. x + 14y = 84
2x – 7y = –7
14. 3x – 2y = 11
1
x– 𝑦=4
2
16. 1 – 3 x – y = 3
17. 4x – 5y = –7
2x + y = 25
y = 5x
3. x = 2y + 7
x=y+4
6. 3x + y = 12
y = –x – 2
9. x – 5y = 36
2x + y = –16
12. 0.3x – 0.2y = 0.5
x – 2y = –5
1
15. 𝑥 + 2y = 12
2
x – 2y = 6
18. x + 3y = –4
2x + 6y = 5
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19. EMPLOYMENT 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.
a. Write a system of equations to represent the situation.
b. What is the total price of the athletic shoes Kenisha needs to sell to earn the same income from each pay scale?
c. Which is the better offer?
20. MOVIE TICKETS Tickets to a movie cost $7.25 for adults and $5.50 for students. A group of friends purchased
8 tickets for $52.75.
a. Write a system of equations to represent the situation.
b. How many adult tickets and student tickets were purchased?
21. BUSINESS Mr. Randolph finds that the supply and demand for gasoline at his station are generally given by the following
equations.
x – y = –2
x + y = 10
Use substitution to find the equilibrium point where the supply and demand lines intersect.
22. GEOMETRY The measures of complementary angles have a sum of 90 degrees. Angle A and angle B are complementary, and
their measures have a difference of 20°. What are the measures of the angles?
23. MONEY Harvey has some $1 bills and some $5 bills. In all, he has 6 bills worth $22. Let x be the number of $1 bills and let y be
the number of $5 bills. Write a
system of equations to represent the information and use substitution to determine how many bills of each
denomination Harvey has.
24. POPULATION Sanjay is researching population trends in South America. He found that the population of Ecuador to increased
by 1,000,000 and the population of Chile to increased by 600,000 from 2004 to 2009. The table displays the information he found.
Ecuador
13,000,000
5-Year
Population
Change
+1,000,000
Chile
16,000,000
+600,000
Country
2004
Population
Source: World Almanac
If the population growth for each country continues at the same rate, in what year are the populations of Ecuador and Chile
predicted to be equal?
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Algebra 1
Section 6.3 Notes: Elimination Using Addition and Subtraction
You have learned about solving a system of equations using the graphing method and the substitution method. A third way to solve a
system of equations is called elimination. Elimination involves using addition or subtraction to solve a system.
Example 1: Use elimination to solve the system of equations.
a)
–3x + 4y = 12
3x – 6y = 18
b) 3x – 5y = 1
2x + 5y = 9
Example 2: Four times one number minus three times another number is 12. Two times the first number added to three times the
second number is 6. Write a system of linear equations and then use elimination to solve it and find the numbers.
Example 3: Use elimination to solve the system of equations.
a)
4x + 2y = 28
4x – 3y = 18
b) 9x – 2y = 30
x – 2y = 14
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Example 4:
a) A hardware store earned $956.50 from renting ladders and power tools last week. The store charged 36 days for ladders
and 85 days for power tools. This week the store charged 36 days for ladders, 70 days for power tools, and earned $829.
How much does the store charge per day for ladders and for power tools?
b)
For a school fundraiser, Marcus and Anisa participated in a walk-a-thon. In the morning, Marcus walked 11 miles and
Anisa walked 13. Together they raised $523.50. After lunch, Marcus walked 14 miles and Anisa walked 13. In the
afternoon they raised $586.50. How much did each raise per mile of the walk-a-thon?
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Algebra 1
6.3 Worksheet
Use elimination to solve each system of equations.
1. x – y = 1
2. p + q = –2
3. 4x + y = 23
x + y = –9
p–q=8
4. 2x + 5y = –3
5. 3x + 2y = –1
6. 5x + 3y = 22
2x + 2y = 6
4x + 2y = –6
5x – 2y = 2
7. 5x + 2y = 7
8. 3x – 9y = –12
9. –4c – 2d = –2
3x – 15y = –6
2c – 2d = –14
–2x + 2y = –14
10. 2x – 6y = 6
11. 7x + 2y = 2
2x + 3y = 24
7x – 2y = –30
13. 2x + 4y = 10
14. 2.5x + y = 10.7
x – 4y = –2.5
16. 4a + b = 2
4a + 3b = 10
2.5x + 2y = 12.9
1
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17. – x – = –2
1
3
3
2
3
x– 𝑦 =4
3
3x – y = 12
12. 4.25x – 1.28y = –9.2
x + 1.28y = 17.6
15. 6m – 8n = 3
2m – 8n = –3
3
1
18. x – 𝑦 = 8
4
3
2
2
1
x – 𝑦 = 19
2
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19. The sum of two numbers is 41 and their difference is 5. What are the numbers?
20. Four times one number added to another number is 36. Three times the first number minus the other number is 20. Find the
numbers.
21. One number added to three times another number is 24. Five times the first number added to three times the other number is 36.
Find the numbers.
22. LANGUAGES English is spoken as the first or primary language in 78 more countries than Farsi is spoken as the first
language. Together, English and Farsi are spoken as a first language in 130 countries. In how many countries is English spoken
as the first language? In how many countries is Farsi spoken as the first language?
23. DISCOUNTS At a sale on winter clothing, Cody bought two pairs of gloves and four hats for $43.00. Tori bought two pairs of
gloves and two hats for $30.00. What were the prices for the gloves and hats?
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Algebra 1
Section 6.4 Notes: Elimination Using Multiplication
Two systems don’t always have to have the same or opposite coefficients for a variable to use elimination. You can use multiplication
and elimination to solve a system when this is the case.
Example 1: Use elimination to solve the system of equations.
a)
2x + y = 23
3x + 2y = 37
b) x + 7y = 12
3x – 5y = 10
Example 2: Use elimination to solve the system of equations.
a)
4x + 3y = 8
3x – 5y = –23
b) 3x + 2y = 10
2x + 5y = 3
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Example 3:
a) A fishing boat travels 10 miles downstream in 30 minutes. The return trip takes the boat 40 minutes. Find the rate in
miles per hour of the boat in still water.
b)
A helicopter travels 360 miles with the wind in 3 hours. The return trip against the wind takes the helicopter 4 hours.
Find the rate of the helicopter in still air.
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Algebra 1
6.4 Worksheet
Use elimination to solve each system of equations.
1. 2x – y = –1
3x – 2y = 1
4. 2x – 4y = –22
3x + 3y = 30
7. 3x + 4y = 27
5x – 3y = 16
10. 6x – 3y = 21
2x + 2y = 22
2. 5x – 2y = –10
3x + 6y = 66
5. 3x + 2y = –9
5x – 3y = 4
8. 0.5x + 0.5y = –2
x – 0.25y = 6
11. 3x + 2y = 11
2x + 6y = –2
3. 7x + 4y = –4
5x + 8y = 28
6. 4x – 2y = 32
–3x – 5y = –11
3
9. 2x – 𝑦 = –7
4
1
x+ 𝑦=0
2
12. –3x + 2y = –15
2x – 4y = 26
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13. Eight times a number plus five times another number is –13. The sum of the two numbers is 1. What are the numbers?
14. Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7.
Find the numbers.
15. FINANCE Gunther invested $10,000 in two mutual funds. One of the funds rose 6% in one year, and the other rose 9% in one
year. If Gunther’s investment rose a total of $684 in one year, how much did he invest in each mutual fund?
16. CANOEING Laura and Brent paddled a canoe 6 miles upstream in four hours. The return trip took three hours. Find the rate at
which Laura and Brent paddled the canoe in still water.
17. NUMBER THEORY The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 45 more
than the original number. Find the number.
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Algebra 1
Section 6.5 Notes: Applying Systems of Linear Equations
You have learned five methods for solving systems of linear equations. The table summarizes the methods and the types of systems
for which each method works best.
Example 1:
a) Determine the best method to solve the system of equations. Then solve the system.
2x + 3y = 23
4x + 2y = 34
b) POOL PARTY At the school pool party, Mr. Lewis bought 1 adult ticket and 2 child tickets for $10.
Mrs. Vroom bought 2 adult tickets and 3 child tickets for $17. Write a system of linear equations for this situation and
then determine the best method to solve the system of equations. Then solve the system.
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Example 2:
a)
CAR RENTAL Ace Car Rental rents a car for $45 and $0.25 per mile. Star Car Rental rents a car for $35 and $0.30 per
mile. How many miles would a driver need to drive before the cost of renting a car at Ace Car Rental and renting a car at Star
Car Rental were the same?
b)
VIDEO GAMES The cost to rent a video game from Action Video is $2 plus $0.50 per day. The cost to rent a video game
at TeeVee Rentals is $1 plus $0.75 per day. After how many days will the cost of renting a video game at Action Video be
the same as the cost of renting a video game at TeeVee Rentals?
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Algebra 1
6.5 Worksheet
Determine the best method to solve each system of equations. Then solve the system.
1. 5x + 3y = 16
3x – 5y = –4
3. y = 3x – 24
2. 3x – 5y = 7
2x + 5y = 13
4. –11x – 10y = 17
5x – y = 8
5x – 7y = 50
5. 4x + y = 24
6. 6x – y = –145
5x – y = 12
x = 4 – 2y
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7. VEGETABLE STAND A roadside vegetable stand sells pumpkins for $5 each and squashes for $3 each. One day they sold 6 more
squash than pumpkins, and their sales totaled $98. Write and solve a system of equations to find how many pumpkins and quash
they sold?
8. INCOME Ramiro earns $20 per hour during the week and $30 per hour for overtime on the weekends. One week Ramiro earned a
total of $650. He worked 5 times as many hours during the week as he did on the weekend. Write
and solve a system of equations to determine how many hours of overtime Ramiro worked on the weekend.
9. BASKETBALL Anya makes 14 baskets during her game. Some of these baskets were worth 2-points and others were worth 3points. In total, she scored 30 points. Write and solve a system of equations to find how 2-points baskets she made.
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Algebra 1
Section 6.6 Notes: Systems of Inequalities
A set of two or more inequalities with the same variable is called a system of inequalities.
The solution of a system of inequalities with two variables is the set of ordered pairs that satisfy all of the inequalities in the system.
The solution set is represented by the overlap, or intersection, of the graphs of the inequalities.
Example 1:
a) Solve the system of inequalities by graphing.
y < 2x + 2
y≥–x–3
b)
Choose the correct solution to the system: 2x + y ≥ 4 and x + 2y > –4.
A.
C.
B.
D.
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Example 2: Solve the system of inequalities by graphing.
y ≥ –3x + 1
y ≤ –3x – 2
Example 3:
a) SERVICE A college service organization requires that its members maintain at least a
3.0 grade point average, and volunteer at least
10 hours a week. Define the variables and write a system of inequalities to represent this situation. Then graph the system.
b) SERVICE A college service organization requires that its members maintain at least a
3.0 grade point average, and volunteer at least 10 hours a week. Name one possible solution.
\
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Algebra 1
6.6 Worksheet
Solve each system of inequalities by graphing.
1. y > x – 2
y≤x
2. y ≥ x + 2
y > 2x + 3
3. x + y ≥ 1
x + 2y > 1
4. y < 2x – 1
y>2–x
5. y > x – 4
2x + y ≤ 2
6. 2x – y ≥ 2
x – 2y ≥ 2
7. FITNESS Diego started an exercise program in which each week he works out at the
gym between 4.5 and 6 hours and walks between 9 and 12 miles.
a. Make a graph to show the number of hours Diego works out at the gym and the
number of miles he walks per week.
b. List three possible combinations of working out and walking that meet Diego’s goals.
8. SOUVENIRS 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.
a. Make a graph showing the numbers of each price of stone Emily can purchase.
b. List three possible solutions.
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