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Section 1.1
Introduction to
Algebra:
Variables and
Mathematical
Models
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Variables in Algebra
Algebra uses letters such as x and y to
represent numbers. If a letter is used to
represent various numbers, it is called a
variable. For example, the variable x might
represent the number of minutes you can lie
in the sun without burning when you are not
wearing sunscreen.
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Variables in Algebra
Suppose you are wearing number 6
sunscreen. If you can normally lie in the sun
x minutes without burning, with the number 6
sunscreen, you can lie in the sun 6 times as
long without burning - that is, 6 times x or 6x
would represent your exposure time without
burning.
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Algebraic Expressions
A combination of variables and numbers
using the operations of addition, subtraction,
multiplication, or division, as well as powers
or roots, is called an algebraic expression.
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Objective 1
Evaluate algebraic expressions.
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Order of Operations - PEMDAS
Order of Operations
1. Perform all operations within grouping symbols,
such as parentheses.
2. Do all multiplications in the order in which they
occur from left to right.
3. Do all additions and subtractions in the order in
which they occur from left to right.
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Example
Evaluate each algebraic expression for x  2.
a. 5  3 x
5  3(2)
56
11
b. 5( x  7)
5(2  7)
5(9)
45
Replace the x with 2.
Perform the multiplication.
Perform the addition.
Replace the x with 2
Perform the addition.
Perform the multiplication.
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Objective 1: Example
1a. Evaluate the expression 2(x  6) for x  10.
x
2(x  6)  2(10  6)
 2(16)
 32
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Objective 1: Example
6x  y
1b. Evaluate the expression
2y  x  8
for x  3 and y  8.
y
x
63  8
6x  y

2y  x  8 2  8  3  8
y
x
18  8

16  3  8
10

5
2
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Objective 2
Translate English phrases into algebraic
expressions.
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Translating Phrases into Expressions
English Phrase
sum
plus
increased by
more than
difference
minus
decreased by
less than
Mathematical Operation
Addition
Subtraction
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Translating Phrases into Expressions
English Phrase
product
times
of (used with fractions)
twice
quotient
divide
per
ratio
Mathematical Operation
Multiplication
Division
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Objective 2: Example
Write each English phrase as an algebraic
expression. Let the variable x represent the
number.
2a.
the product of 6 and a number
6x
2b.
a number added to 4
4x
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Objective 2: Example (cont)
2c.
three times a number, increased by 5
3x  5
2d.
twice a number subtracted from 12
12  2 x
2e.
the quotient of 15 and a number
15
x
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Objective 3
Determine whether a number is a solution of
an equation.
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Equations
An equation is a statement that two
algebraic expressions are equal. An
equation always contains the equality
symbol  . Some examples of equations
are:
5x  2  15
3x  7  2x
3( z  1)  4( z  7)
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Equations
Solutions of equations are values of the variable
that make the equation a true statement. To
determine whether a number is a solution, substitute
that number for the variable and evaluate both sides
of the equation. If the values on both sides of the
equation are the same, the number is a solution.
For example, 2 is a solution of x  4  3x
since when we substitute the 2 for x, we get 2  4  3(2)
or equivalently, 6  6
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Objective 3: Example
3a. Determine whether the given number is a
solution of the equation.
9 x  3  42; 6
9 x  3  42
9(6)  3  42
54  3  42
51  42, false
6 is not a solution.
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Objective 3: Example
3b. Determine whether the given number is a
solution of the equation.
2( y  3)  5 y  3; 3
2( y  3)  5 y  3
2(3  3)  5(3)  3
2(6)  15  3
12  12, true
3 is a solution.
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Objective 4
Translate English sentences into algebraic
equations.
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Translate English sentences into Equations
Earlier in the section, we translated English
phrases into algebraic expressions. Now we will
translate English sentences into equations. You’ll
find that there are a number of different words and
phrases for an equation’s equality symbol.
equals
gives
yields
is the same as
is/was/will be
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Objective 4: Example
4a.
Write the sentence as an equation. Let the
variable x represent the number.
The quotient of a number and 6 is 5.
x
5
6
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Objective 4: Example
4b.
Write the sentence as an equation. Let the
variable x represent the number. Seven
decreased by twice a number yields 1.
7  2x  1
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Objective 5
Evaluate formulas.
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Formulas and Mathematical Models
One aim of algebra is to provide a compact,
symbolic description of the world. A formula
is an equation that expresses a relationship
between two or more variables.
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Formulas and Mathematical Models
One variety of crickets chirps faster as the
temperature rises. You can calculate the
temperature by applying the following
formula:
T  0.3n  40
If you are sitting on your porch and hear 50
chirps per minute, then the temperature is:
T  0.3(50)  40  15  40  55 degrees
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Formulas and Mathematical Models
The process of finding formulas to describe
real-world phenomena is called
mathematical modeling. Formulas together
with the meaning assigned to the variables
are called mathematical models.
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Formulas and Mathematical Models
In creating mathematical models, we strive
for both simplicity and accuracy. For
example, the cricket formula is relatively
easy to use. But you should not get upset if
you count 50 chirps per minute and the
temperature is 53 degrees rather than 55.
Many mathematical formulas give an
approximate rather than exact description
of the relationship between variables.
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Objective 5: Example
5.
Divorce rates are considerably higher for couples
who marry in their teens. The line graphs in the
figure show the percentages of marriages ending
in divorce based on the wife’s age at marriage.
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Objective 5: Example (cont)
Here are two mathematical models that
approximate the data displayed by the line
graphs.
Wife is under 18 at time of marriage:
d  4n  5
Wife is over 25 at time of marriage:
d  2.3n  1.5
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Objective 5: Example (cont)
a. Use the appropriate formula to determine
the percentage of marriages ending in
divorce after 15 years when the wife is
under 18 at the time of marriage.
d  4n  5
d  4(15)  5  65
65% of marriages end in divorce after 15
years when the wife is under 18 at the
time of marriage.
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Objective 5: Example (cont)
b. Use the appropriate line graph in the
figure to determine the percentage of
marriages ending in divorce after 15
years when the wife is under 18 at the
time of marriage.
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Objective 5: Example (cont)
According to the line graph, 60% of
marriages end in divorce after 15 years
when the wife is under 18 at the time of
marriage.
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Objective 5: Example (cont)
c. Does the value given by the mathematical
model underestimate or overestimate the
actual percentage of marriages ending in
divorce after 15 years as shown by the
graph? By how much?
The mathematical model overestimates
the actual percentage shown in the graph
by 5%.
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