Download Equations in Two Variables Practice and Problem Solving

Name _______________________________________ Date __________________ Class __________________
For each equation, find the ordered pair whose x-coordinate is 4.
1. y  18  5x
2. 2x  3y  4
_______________________
3. 7x  6y  10
________________________
________________________
For each equation, find the ordered pairs whose x-coordinates
are 3 or 3.
4. x  2y  1
_______________________
5.
y
x

1
3
3
6. 3x  y  10  0
________________________
________________________
For each equation, find y for x  2, 1, 0, 1, and 2. List the ordered
pairs and then graph the equation on the coordinate grid given.
7. y  2x  3
8. x  y  4
_______________________________________
________________________________________
9. y  3x  1
10. x  2y 10  0
_______________________________________
________________________________________
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76
Name _______________________________________ Date __________________ Class __________________
LESSON
5-1
Equations in Two Variables
Practice and Problem Solving: C
For each equation, find the ordered pair whose x-coordinate is
1. 2x  5y  3
2.
_______________________
8x + y
= 2
x - 8y
1
.
2
3. y  x 2  3 x  1
________________________
________________________
For each equation, find the ordered pairs whose x-coordinates are 5
or 5.
4. 4x  6y  8
_______________________
5.
y -4
x
= 2
7
x -4
6. x 2 + y 2 = 169
________________________
________________________
For each equation, choose three ordered pairs to graph. List the
ordered pairs and then graph the equation on the coordinate grid
given. Indicate the scales you use on the x- and y-axes.
7. 3x  2y  10
8. 2y  5x  20
_______________________________________
________________________________________
9. x  y  2(x  y)
10. y  8(2x  1)
_______________________________________
________________________________________
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77
Name _______________________________________ Date __________________ Class __________________
Equations in Two Variables
LESSON
5-1
Practice and Problem Solving: D
Tell whether each ordered pair is a solution to 2x  5y  6. The first
one is done for you.
1. (3, 0)
3. (0, 1)
2. (8, 2)
yes
_______________
_______________
4. (2, 2)
_______________
________________
Tell whether each ordered pair is a solution to 2x  y2  40. The first
one is done for you.
5. (10, 15)
6. (2, 6)
no
_______________
7. (12, 8)
_______________
8. (4, 12)
_______________
________________
Complete the table of values to find solutions for each equation.
The first one is done for you.
9.
10.
x
3x  y
(x, y)
2
6
(2, 6)
2
1
3
(1, 3)
1
0
0
(0, 0)
0
1
3
(1, 3)
1
2
6
(2, 6)
2
x
y  x  4
(x, y)
Graph the ordered pairs from the problem indicated. Connect your
points with a line. The first one is done for you.
11. Problem 9
12. Problem 10
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78
Name _______________________________________ Date __________________ Class __________________
Equations in Two Variables
LESSON
5-1
Reteach
Solutions to equations with two variables, x and y, are ordered pairs.
An ordered pair is a solution to an equation if the equation is true when you
substitute the first coordinate for x and the second coordinate for y. For
example, the ordered pair (3, 5) is a solution to x  y  8 because 3  5  8.
The equation x  y  8 has infinitely many solutions, or ordered pairs, that make it true.
You can use a table to represent some of the ordered pairs in the solution of x  y  8.
x
1
0
2
4
5
8
9
y
9
8
6
4
3
0
1
You can picture all of the solutions to x  y  8 by drawing the graph of this equation.
The graph of an equation contains every pair of points that make the equation true.
To graph an equation, find at least three pairs of values for x and y that make the equation
true. Plot those ordered pairs and join them with a straight line that extends both directions.

Example
Complete the table and graph this equation: y  4  2x
x
3
2
1
0
y
2
0
2
4
Complete each table and graph each equation.
1. 4y  3x  12
x
y
2
0
2. y  x  3
2
4
x
y
3
3. 2y  x  4
1
1
0
x
y
2
0
2
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79
4
Name _______________________________________ Date __________________ Class __________________
LESSON
5-1
Equations in Two Variables
Reading Strategies: Understanding Relationships
The table shows the number of cartons and the number of eggs that
are in each carton.
Number of Cartons
Number of Eggs
1
12
2
24
3
36
4
48
5
60
6
72
1. How many eggs are there in one carton? _______________________________________
2. How many eggs are there in 4 cartons? ________________________________________
3. If you had 36 eggs, how many cartons would that be? _____________________________
You can show the relationship between the number of cartons and the
number of eggs as an ordered pair. For example: (1, 12) is the ordered pair
which stands for 1 egg carton, 12 eggs. An ordered pair has two numbers,
and they must be placed in the correct order.
4. Write the ordered pair for 3 egg cartons and 36 eggs. ______________________________
5. Write the ordered pair for 5 egg cartons and 60 eggs. ______________________________
Write “true” or “false.”
6. The ordered pair for 6 egg cartons and 72 eggs is (72, 6).
________________________________________________________________________________________
7. The ordered pair for 2 egg cartons and 24 eggs is (2, 24).
________________________________________________________________________________________
Write ordered pairs for each of the following.
8. One deck of cards contains 52 cards. ___________________________________________________
9. Two packs of soda contain 12 cans. _____________________________________________________
10. Three packages of gum contain 15 pieces. ______________________________________________
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80
Name _______________________________________ Date __________________ Class __________________
LESSON
5-1
Equations in Two Variables
Success for English Learners
Problem 1
How many variables are in the equation?
y  20  10x There are 2 variables.
For equations with two variables, the solution can be written as an
ordered pair, like (x, y).
Problem 2
Is (2, 5) a solution of the equation y  3x  2?
Substitute for x.
Substitute for y.
(2,5)
y  3x  2
5  3(2)  2
58
Problem 3
(2, 5) is
not a
solution.
Be careful when reading a solution.
For the equation c  1.06p, a solution is (160, 169.60). This equation
represents the total cost after sales tax.
The value of p is 160.
The value of c is 169.60.
1. In Problem 1, write the solution as an ordered pair if x  2. ____________
2. Is (10, 4) a solution to the equation in Problem 2? ___________________
3. In Problem 3, if p  1, then c  1.06. Is the ordered pair
(1.06, 1) a reasonable solution to the question?
________________________________________________________________________________________
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81
Name _______________________________________ Date __________________ Class __________________
LESSON
5-2
Representing Functions
Practice and Problem Solving: A/B
Express each relation as a table, as a graph, and as a mapping
diagram.
1. {(2, 5), (1, 1), (3, 1), (1, 2)}
x
y
2. {(5, 3), (4, 3), (3, 3), (2, 3), (1, 3)}
x
y
Give the domain and range of each relation. Tell whether the relation
is a function. Explain.
3.
4.
5.
x
y
1
4
2
5
0
6
1
7
2
8
D: ____________________
D: ____________________
D: ____________________
R: ____________________
R: ____________________
R: ____________________
Function? ______________
Function? _____________
Function?_____________
Explain: ______________
Explain: ______________
Explain: _____________
________________________
________________________
________________________
________________________
________________________
________________________
________________________
________________________
________________________
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82
Name _______________________________________ Date __________________ Class __________________
LESSON
5-2
Representing Functions
Practice and Problem Solving: C
Graph each relation. Then explain whether it is a function or not.
1. {(1, 2), (2, 2), (3, 3), (4, 3)}
2. {(1, 5), (2, 4), (3, 5), (3, 4), (4, 4), (5, 5)}
_______________________________________
________________________________________
_______________________________________
________________________________________
Solve.
3. In the relation y  x2, y is a function of x but x is not a function of y.
Explain why.
________________________________________________________________________________________
________________________________________________________________________________________
4. Find the domain and range of f ( x )  x 2  8. Explain your reasoning.
________________________________________________________________________________________
________________________________________________________________________________________
5. Find the domain and range of f (x) =
1
. Explain your reasoning.
x-1
________________________________________________________________________________________
________________________________________________________________________________________
6. The function INT(x) takes any x and rounds it down to the nearest
integer. INT(x) is used in spreadsheet programs. Find INT(x) for
x  4.6,  2.3, and 2 . Then find the domain and range.
________________________________________________________________________________________
________________________________________________________________________________________
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83
Name _______________________________________ Date __________________ Class __________________
LESSON
5-2
Representing Functions
Practice and Problem Solving: D
Identify the domain and range for each set of ordered pairs. The first
one is started for you.
1. {(0, 1), (3, 1), (5, 1)}
2. {(2, 2), (3, 4), (1, 2), (3, 4), (0, 5)}
Domain: {0, 3, 5}
_______________________________________
________________________________________
_______________________________________
________________________________________
State whether each mapping diagram shows a function. If not, explain
why. The first one is done for you.
3.
4.
It is not a function because
_______________________________________
________________________________________
_______________________________________
9 is paired with two outputs.
_______________________________________ `
5.
6.
_______________________________________
________________________________________
_______________________________________
________________________________________
A club’s president kept track of membership over its first seven
years. Use her graph below to solve Problems 7–9. The first one is
done for you.
7. What is the range in the graph?
{30,
40, 50, 60}
_______________________________________
8. What is the domain in the graph?
_______________________________________
9. Does the graph show a function? Explain
your reasoning.
_______________________________________
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84
Name _______________________________________ Date __________________ Class __________________
Representing Functions
LESSON
5-2
Reteach
An equation such as y  3x  2 defines a function. If you choose a value
for x, you can then calculate a corresponding value for y. If the equation
defines a function, then each value for x will correspond with only one
value for y. Because each value for y depends on the chosen value for x,
an equation that is a function can be written in function notation, such as
f(x)  3x  2.
In function notation, the f next to the (x) does NOT mean to multiply.
Say this as “the function of x is 3x  2” or, “f of x equals 3x  2.”

Example
To graph f(x)  3x  2, make a table of three or
more values for x, and find the corresponding values for y.
Then plot the ordered pairs and
join them with a line that continues in both directions.
x
2
1
0
1
f (x)
4
1
2
5
Complete the table and graph the function f(x)  2x  4.
1.
x
3
2
1
0
f (x)
The domain of a function is the set of values for x.
The range of a function is the set of values for y.
In the function f(x)  2x  4, both the domain and the range are all real numbers.
Example
Does this table describe a function? Explain.
If it is a function, give the domain and range.
Yes, because each x is paired with only one y.
The domain is {1, 2, 3, 4}, and the range is {3, 4, 5}.
x
1
2
3
4
f (x)
3
3
4
5
Tell whether each pairing of numbers describes a function.
If so, write the function and give the domain and range.
2.
x
0
1
2
3
f (x)
0
1
2
3
_______________________
3.
x
0
1
1
4
f (x)
0
1
1
2
________________________
4.
x
0
1
2
3
f (x)
1
2
3
4
________________________
________________________________________________________________________
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85
Name _______________________________________ Date __________________ Class __________________
LESSON
5-2
Representing Functions
Reading Strategies: Use Examples and Non-Examples
A function is defined as a relation that pairs each domain value with exactly
one range value. No x-value can be repeated with a different y-value.
Examples
Non-Examples
{(1, 3), (2, 4), (3, 5), (6, 8)}
{(1, 3), (2, 4), (3, 5), (3, 8)}
x
2
1
0
1
2
x
6
6
6
6
6
y
6
6
6
6
6
y
2
1
0
1
2
Answer the following.
1. Give your own example of a function
in table form.
2. Give an example of a mapping diagram
that is NOT a function.
x
y
3. Explain why the relation in problem 2 is not a function.
________________________________________________________________________________________
________________________________________________________________________________________
Tell whether each of the following is a function by writing yes or no.
4. ____________
5. __________
x
y
3
8
2
5
1
1
1
4
1
6
6. __________
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86
Name _______________________________________ Date __________________ Class __________________
LESSON
5-2
Representing Functions
Success for English Learners
Problem 1
DOMAIN
RANGE
Which x-coordinates are represented in
the graph?
Which y-coordinates are represented in
the graph?
All of the x-coordinates including and
between 1 and 3 are represented on the
graph. So, the Domain is 1  x  3.
All of the y-coordinates including and
between 2 and 4 are represented on the
graph. So, the Range is 2  y  4.
Problem 2
FUNCTION
Each x-coordinate can only have ONE y-coordinate.
Function
NOT a Function
(75, 2), (68, 2), (125, 3)
(7, 0), (7, 1), (9, 7), (12, 1), (15, 0)
x
y
75
2
68
2
125
3
Each x-coordinate
has only ONE
y-coordinate.
x
y
7
0
7
1
9
7
12
1
15
0
The x-coordinate 7
has TWO
y-coordinates.
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87
Name ________________________________________ Date __________________
Class _________________
1. How can you change the relation that is not a function in
Problem 2 so that it is a function?
_______________________________________________________________________________________
2. If the domain of a function is given as 4  x  4 is 5 part of the domain? Why?
_______________________________________________________________________________________
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instructor.
71
Name ________________________________________ Date __________________
LESSON
5-3
Class _________________
Sequences
Practice and Problem Solving: A/B
Find the first four terms of each sequence.
2
2. f (n)  n  2n  5
1. f (n)  3n  1
_______________________________________
3. f (n ) 
_______________________________________
n6
2n  3
4. f (n )  n  1
_______________________________________
_______________________________________
5. f (n )  (n  1)(n  2)
n2
6. f (1)  3, f (n )  f (n  1)  7 for
_______________________________________
_______________________________________
7. f (1)  9, f (n )  2f (n  1)  1 for n  2
8. f (n) 
_______________________________________
9. f (1)  6, f (n )  
n(n  1)
2
_______________________________________
1
for n  2
f (n  1)
10. f (1)  16, f (n )  f (n  1) for n  2
_______________________________________
_______________________________________
A ferry charges $40 for each car and $8.50 for each person in
the car. Use this information for Problems 11–16.
11. Find the cost for a car containing only a driver.
_______________________________________________________________________________________
12. Find the cost for a car containing four people.
_______________________________________________________________________________________
13. Write an explicit rule for this situation.
_______________________________________________________________________________________
14. Write a recursive rule for this situation.
_______________________________________________________________________________________
15. A car is charged $91. Determine how many people are in the
car.
_______________________________________________________________________________________
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instructor.
72
Name ________________________________________ Date __________________
Class _________________
16. Explain whether this situation represents a function or not.
_______________________________________________________________________________________
_______________________________________________________________________________________
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instructor.
73
Name ________________________________________ Date __________________
LESSON
5-3
Class _________________
Sequences
Practice and Problem Solving: C
Find the first four terms of each sequence.
1. f (n)  n3  n2  1
2. f (n ) 
_______________________________________
1
1

n n 1
_______________________________________
3. f (1)  3, f (n )  5f (n  1)  1 for n  2
4. f (n ) 
_______________________________________
n(n  1)(2n  1)
6
_______________________________________
5. f (1)  9, f (n )  13  f (n  1) for n  2
6. f (n ) 
_______________________________________
n2  1
n2  1
_______________________________________
8. f (1)  1, f (n )  n 
7. f (1)  3, f (n)  n2  2 f (n  1) for n  2
1
for
f (n  1)
n2
_______________________________________
_______________________________________
9. f (1)  2, f (n)  1  (1)n f (n  1) for n  2
10. f (n ) 
_______________________________________
n 2

12 3
_______________________________________
Solve.
11. In the Fibonacci sequence, f (1)  1 , f (2)  1 , and
f (n )  f (n  2)  f (n  1) for n  3 . Find the first 10 terms of the
Fibonacci sequence.
_______________________________________________________________________________________
12. Use f (n ) from Problem 11 to create a new sequence:
f (n )
r (n ) 
. Write the first eight terms of this sequence as
f (n  1)
decimals. If necessary, round a term to three decimal places.
Explain any patterns you see.
_______________________________________________________________________________________
_______________________________________________________________________________________
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instructor.
74
Name ________________________________________ Date __________________
Class _________________
(n  1)(n  2)
.
2
Then study those terms to find a recursive formula for the
sequence.
13. Find the first six terms of the sequence f (n ) 
_______________________________________________________________________________________
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instructor.
75
Name ________________________________________ Date __________________
LESSON
5-3
Class _________________
Sequences
Practice and Problem Solving: D
Find the first four terms of each sequence. The first one is done for you.
1. f (n )  n  5
2. f ( n )  4n
6,_______________________________________
7, 8, 9
_______________________________________
4. f (n ) 
3. f (n )  n  2
_______________________________________
n
6
_______________________________________
5. f (n )  8  n
6. f (n )  6n  1
_______________________________________
_______________________________________
7. f (n )  5n
8. f (n )  1  4n
_______________________________________
_______________________________________
9. f (1)  10, f (n )  4f (n  1) for n  2
n2
10. f (1)  2, f (n )  f (n  1)  21 for
_______________________________________
_______________________________________
A limousine service charges $120 per day plus $5 for each mile driven.
Use this information for Problems 11–16. The first one is done for you.
11. Find the cost for using this limousine to ride 20 miles.
$220
_______________________________________________________________________________________
12. Find the cost for using this limousine to ride 100 miles.
_______________________________________________________________________________________
13. Write an explicit rule for this situation.
_______________________________________________________________________________________
14. Find the cost for using this limousine to ride 1 mile.
_______________________________________________________________________________________
15. If you were riding in this limousine, how much would one extra mile cost?
_______________________________________________________________________________________
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instructor.
76
Name ________________________________________ Date __________________
Class _________________
16. Write a recursive rule for this situation.
_______________________________________________________________________________________
LESSON
5-3
Sequences
Reteach
A list of numbers in a specific order, or pattern, is called a sequence. Each number, or term,
in the sequence corresponds with the position number that locates it in the list. For example,
in the sequence 2, 4, 6, 8.., the first term is 2, the second term is 4, and the third term is 6.
You can write a sequence as a function, where the domain is {1, 2, 3, 4,…} or the set of
position numbers. The range is the set of the numbers, or terms, in the list.
Domain or position number: n
1
2
3
4
5
Range or term: f(n)
2
4
6
8
10
This sequence can be described by an explicit rule that defines each f(n) in terms of n.
For the table above, the explicit rule is f(n)  2n.
Using this rule, calculate the 11th term (n  11) of this sequence: f(11)  2(11) or 22.

Example
Complete this table by finding the first 4 terms of the sequence defined by f(n)  n(n 1).
Then find the 10th term of the sequence.
n
1
2
3
4
…10
f(n)
1(1  1)  2
2(2  1)  6
3(3  1)  12
4(4  1)  20
10(10 1)  110
Complete each table for the given sequence.
1. f(n)  3n  2
n
f(n)
1
2
2. f(n) 
3
4
n
1
n1
2
1
2
f(n)
3. f(n)  n  1
3
4
n
1
2
f(n)
In a sequence, the term before the nth term can be written as f(n  1). Sometimes
a sequence is described by giving the first term, then the rule defines each term after
the first one by using the term before. A rule that does this is called a recursive rule.
Example
Write the first four terms of the sequence with f(1)  3 and f(n)  f(n  1)  5 for n  2.
3, (3)  5, (3  5)  5, (3  5  5)  5 which simplifies to 3, 8, 13, 18.
Write the first four terms of each sequence.
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77
3
4
Name ________________________________________ Date __________________
4. f(1)  10 and f(n)  f(n  1)  2 for n  2
n2
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5. f(1)  4 and f(n)  f(n  1)  3 for
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instructor.
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LESSON
5-3
Class _________________
Sequences
Reading Strategies: Follow a Procedure
Using a Recursive Rule to generate a sequence requires knowing each term
in order, then using that term to find the next.
Find the first five terms in the sequence f(n)  2f(n  1)  3 and f(1)  7.
Step 1: Identify the first term and the recursive rule.
The first term is 7, and the recursive rule is 2f(n  1)  3.
n  1 means to use the
previous term in the
sequence, which is why
having f(1) is necessary.
n alone stands for the place.
Step 2: Plug the previous term into the rule, then solve.
Since the first term is 7, plug that into the rule as follows:
f (2)  2f (2  1)  3
 2f (1)  3
 2(7)  3
 14  3
 11
Step 3: Repeat Step 2 until all terms are found.
f (3)  2f (3  1)  3
f (4)  2f (4  1)  3
 2f (2)  3
 2f (3)  3
 2(11)  3
 2(19)  3
 22  3
 38  3
 19
 35
f (5)  2f (5  1)  3
 2f (4)  3
 2(35)  3
 70  3
 67
So, the first five terms of the recursive sequence are 7, 11, 19, 35, and 67.
Using the process illustrated, find the first five terms in the sequences.
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Name ________________________________________ Date __________________
2 f (n )  f (n  1)  7; f (1)  5
1. f (n)  f (n  1)  3; f (1)  7
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1
2
4. f (n )  3f (n  1)  1; f (1)  1
3. f (n )  f (n  1)  4; f (1)  2
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LESSON
5-3
Class _________________
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Sequences
Success for English Learners
Problem 1
Use the explicit rule f(n)  4n to find the first five terms in the
sequence, where n represents the position of the term.
f(1)  4(1)  4
sequence.
n.
f(2)  4(2)  8
f(1) indicates that it is the first term in the
The placement of the term is plugged into the
f(3)  4(3)  12
Once 3
f(4)  4(4)  16
f(3) indicates the third term of the sequence.
is plugged in, solve to find the term of 12.
f(5)  4(5)  20
The first five terms of the sequence are 4, 8, 12, 16, and 20.
Problem 2
Use the explicit rule f(n)  3n  10 to find the first five terms in the
sequence, where n represents the position of the term.
f(1)  3(1)  10  7
sequence.
n.
f(2)  3(2)  10  4
f(1) indicates that it is the first term in the
The placement of the term is plugged into the
f(3)  3(3)  10  1
Once 3
f(4)  3(4)  10  2
f(3) indicates the first term of the sequence.
is plugged in, solve to find the term of 1.
f(5)  3(5)  10  5
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Name ________________________________________ Date __________________
Class _________________
The first five terms of the sequence are 7, 4, 1, 2, and 5.
1. What does the variable n represent in the explicit rules? Give an example.
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2. What number would be plugged into n to find the 10th term of the sequence? The
25th?
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instructor.
81
Name ________________________________________ Date __________________
Class _________________
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instructor.
82