### 1-1 Using Variables

```1-1 Using
Variables
NCSCOS: 1.02 – Use formulas
and algebraic expressions,
including iterative and
recursive forms, to model and
solve problems.
Obj. 1: Modeling Relationships with
Variables  If you earn an hourly wage of \$6.50

Hours Worked
Pay (dollars)
1
6.50 x 1
2
6.50 x 2
3
6.50 x 3
h
6.50 x h
your pay is the number of hours you
work multiplied by 6.5.
In the table at the bottom left, the
variable h stands for the number of
hours you worked. A variable is a
symbol, usually a letter, that
represents one of more numbers.
The expression 6.50h is an algebraic
expression. An algebraic expression
is a mathematical phrase that can
include numbers, variables, and
operation symbols. Algebraic
expressions are sometimes called
variable expressions.
Ex. 1: Writing an
Algebraic Expression
 Write

an algebraic expression for each phrase:
Seven more than n
+
 “More
to the second number n.

The difference of n and seven
−
 “Difference”
indicates subtraction. Begin with the first
number n. Then subtract the second number 7

The product of seven and n

 “Product”
indicates multiplication. Multiply the first
number 7 by the second number n.

The quotient of n and seven
 “Quotient”

indicates division. Divide the first number n
by the second number 7.
Your turn to write an algebraic
expression:
 The
t
quotient of 4.2 and c
minus 15
 Remember:
.

−
To translate an English phrase
into an algebraic expression, you may
need to define one or more variables first.
Ex. 2: Writing an
Algebraic Expression
 Define
a variable and write an algebraic expression
for each phrase.

Two times a number plus 5
 Relate:
two times a number plus 5
 Define: Let n = the number.
 Write: 2
x
n
+ 5

7 less than three times a number
 Relate:
7 less than three times
 Define: Let a = the number.
 Write: 3
x
a
a number
-
7
a variable and write an algebraic
expression for each phrase:
9
less than a number
 The
 The
sum of twice a number and 31
product of one half of a number and one
third of the same number
Obj. 2: Modeling Relationships with
Equations and Formulas
 You
can use algebraic expressions to write an
equation. An equation is a mathematical
sentence that uses an equal sign. If the equation
is true, then the two expressions on either side of
the equal sign represent the same value. An
equation that contains one or more variables in
an open sentence. In everyday language, the
associated equation.
Ex. 3: Writing an
Equation
 Track
One Media sells all CDs for \$12 each. Write
an equation for the total cost of given number of
CDs.


Relate: The total cost is 12 times the number of
CDs bought.
Define: Let n = the number of CDs bought.
Let c = the total cost.

Write:
 Suppose
c
=
12 x
n
the manager at Track One Media raises
the price of each CD to \$15. Write an equation to
find the cost of n CDs.
 Suppose the manager at Track One Media uses
the equation c = 10.99n. What could this mean?
Ex. 4: Real-World
Problem Solving
 Write


an equation for the data in the table below:
Cost of Purchase
Change from \$20
\$20.00
\$0
\$19.00
\$1.00
\$17.50
\$2.50
\$11.59
\$8.41
Relate: Change equals \$20.00 minus
purchase.
Define: Let c = cost of item purchased.
cost of
Let a = amount of change.

Write:
a
=
\$20.00
-
c
Exercises: Practice
and Problem Solving
 Ex.
1: Practice by Example: Write an algebraic
expression for each phrase.
1. 4 more than p
2. y minus 12
3. 12 minus m
4. The product of c and 15
5. The quotient of n and 8
6. The quotient of 17 and k
7. 23 less than x
8. The sum of v and 3
Exercises: Practice
and Problem Solving Continued
 Ex.
2: Define a variable and write an expression for
each phrase.
9. 2 more than twice a number.
10. A number minus 11
11. 9 minus a number
12. A number divided by 82
13. The product of 5 and a number
14. The sum of 13 and twice a number
15. The quotient of a number and 6
16. The quotient of 11 and a number
Exercises: Practice
and Problem Solving Continued
 Ex.
3: Define variables and write an equation
to model each situation.
17. The total cost is the number of cans times
\$0.70.
18. The perimeter of a square equals 4 times the
length of a side.
19. The total length of rope, in feet, used to put
up tents is 60 times the number of tents.
20. What is the number of slices of pizza left
from an 8-slice pizza after you have eaten
some slices?
Exercises: Practice
and Problem Solving Continued
 Ex.
4: Define variables and write an equation
to model the relationship in each table.
21.
23.
Number of
Workers
Built
1
22.
Number of Tapes
Cost
13
1
\$8.50
2
26
2
\$17.00
3
39
3
\$25.50
4
52
4
\$34.00
Number of Sales
Total Earnings
Number of Hours
Total Pay
5
\$2.00
4
\$32
10
\$4.00
6
\$48
15
\$6.00
8
\$64
20
\$8.00
10
\$80
24.
Write an expression for each phrase.
25.
26.
27.
28.
29.
30.
31.
32.
33.
The sum of 9 and k minus 17
6.7 more than 5 times n
9.85 less than the product of t and 37
The quotient of 3b and 4.5
15 plus the quotient of 60 and w
7 minus the product of v and 3
The product of m and 5, minus the quotient of t
and 7
The sum of the quotient of p and 14 and the
quotient of q and 3
8 minus the product of 9 and r
Write a phrase for
each expression.
34.
q+5
35.
3–t
36.
9n + 1
37.

5
38.
7hb
Define variables
and write an equation to model the
relationship in each table.
39.
Number of Days
Change in Height
(meters)
1
40.
Time (months)
Length (inches)
0.165
1
4.1
2
0.330
2
8.2
3
0.495
3
12.3
4
0.660
4
16.4
Use the table below:
Lawns Mowed
Hours
1
2
3
6
 Does
each statement fit the data in the
table? Explain.
 Hours
worked = lawns mowed x 2
 Hours worked = lawns mowed + 3
Challenge:
 The
table at the right shows
the height of the first
bounce when a ball is
dropped from different
heights.
 Write an equation to
describe the relationship
between the height of the
first bounce and the drop
height.
 Suppose
Drop Height (ft.)
Height of First
Bounce (ft.)
1
1
2
2
1
3
1
4
5
1
2
2
2
1
2
you drop the ball from a window 20 ft.
above the ground. Predict how high the ball will
bounce.
Open-Ended
 Describe
a real-world situation that each
equation could represent. Include a definition for
each variable.
d
= 5t
a
=b+3
c
=
40
ℎ
Standardized Test Prep
Which is an algebraic
expression for “six less
than k”?
1.
a.
b.
2.
6

6
6–k
d. k – 6
Which is an algebraic
expression for “the
product of a and 10”?
a. a + 10
b. a – 10
c. 10a
c.
d.
3.

10
4.
Which is an algebraic
expression for “9 more
than v”?
a. v + 9
b. v– 9
c. 9 – v
d. 9v
A container of milk
contains 64 ounces.
Which equation models
the number n of ounces
remaining after you have
drunk m ounces?
a.  − 64 =
b. 64 −  =
c.  − 64 =
d.  −  = 64
Standardized Test Prep
5.
Which equation models the relationship in the
table if r represents the row number and t
represents the number of tulips?
a.  = 3
Row Number
Number of Tulips
b.

=3
1
3
2
6
3
9
=+3
4
12
d.  = 3
Which is an algebraic expression for “the quotient
of r + 5 and b”?
c.
6.
a.
b.
+5

+5
c.
d.

+5

+5

12.
0.2 + 0.7
0.13 + 0.91
0.6 + 0.75
1.09 + 0.37
0.9 x 0.07
0.58 – 0.49
19.
List four prime numbers between 20 and 50.
7.
8.
9.
10.
11.
13.
14.
15.
16.
17.
18.
0.8 – 0.66
1.32 – 0.39
2 x 0.5
0.69 ÷ 3
0.6 ÷ 0.2
1.21 ÷ 11
```