Download Ch1-Sec 1.2

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1.2 – Slide 1
Chapter 1
The Real Number System
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1.2 – Slide 2
1.2
Variables, Expressions, and
Equations
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1.2 – Slide 3
1.2 Variables, Expressions, and Equations
Objectives
1.
2.
3.
4.
5.
Evaluate algebraic expressions, given values for
the variables.
Translate phrases from words to algebraic
expressions.
Identify solutions of equations.
Translate sentences to equations.
Distinguish between expressions and equations.
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1.2 – Slide 4
1.2 Variables, Expressions, and Equations
A variable is a symbol, usually a letter such as x, y, or z.
Different numbers can replace the variables to form specific
statements.
An algebraic expression is a collection of numbers,
variables, operation symbols, and grouping symbols,
such as parentheses, square brackets, or fraction bars.
x + 5, 2m – 9, 8p2 + 6(p – 2)
Algebraic expressions
In 2m – 9, the 2m means 2 · m, the product of 2 and m;
8p2 represents the product of 8 and p2.
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1.2 – Slide 5
1.2 Variables, Expressions, and Equations
Evaluating Expressions
Example 1
Find the value of each algebraic expression if m = 5 and then if
m = 9.
(a) 8m
8m = 8 · 5
= 40
(b) 3m2
3m2 = 3 · 52
Let m = 5.
8m = 8 · 9
Let m = 9.
Multiply.
= 72
Multiply.
Let m = 5.
3m2 = 3 · 92
Let m = 9.
= 3 · 25
Square 5.
= 3 · 81
Square 9.
= 75
Multiply.
= 243
Multiply.
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1.2 – Slide 6
1.2 Variables, Expressions, and Equations
Evaluating Expressions
CAUTION
In example 1(b), 3m2 means 3 · m2; not 3m · 3m. Unless
parentheses are used, the exponent refers only to the variable or
number just before it. Use parentheses to write 3m · 3m with
exponents as (3m)2.
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1.2 – Slide 7
1.2 Variables, Expressions, and Equations
Evaluating Expressions
Example 2
Find the value of the expression if x = 5 and y = 3.
9 x  8 y 9 5  8  3
(b)

2x  y
2 5  3
45  24

10  3
21

7
3
Replace x with 5 and y with 3.
Multiply.
Subtract.
Divide.
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1.2 – Slide 8
1.2 Variables, Expressions, and Equations
Translating Word Phrases to Algebraic Expressions
PROBLEM-SOLVING HINT
Sometimes variables must be used to change word phrases into
algebraic expressions. This process will be important later for
solving applied problems.
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1.2 – Slide 9
1.2 Variables, Expressions, and Equations
Translating Word Phrases to Algebraic Expressions
Example 3
Write each word phrase as an algebraic expression, using x as
the variable.
(a) The sum of a number and 9
“Sum” is the answer to an addition problem. This phrase
translates as
x + 9, or 9 + x.
(b) 7 minus a number
“Minus” indicates subtraction, so the translation is
7 – x.
Note that x – 7 would not be correct.
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1.2 – Slide 10
1.2 Variables, Expressions, and Equations
Translating Word Phrases to Algebraic Expressions
Example 3 (continued)
Write each word phrase as an algebraic expression, using x as
the variable.
(c) A number subtracted from 12
Since a number is subtracted from 12, write this as
12 – x.
(d) The product of 11 and a number
11 · x, or 11x
(e) 5 divided by a number
5
5  x, or
x
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1.2 – Slide 11
1.2 Variables, Expressions, and Equations
Translating Word Phrases to Algebraic Expressions
Example 3 (concluded)
Write each word phrase as an algebraic expression, using x as
the variable.
(f) The product of 2 and the difference between a number
and 8.
2 (x – 8)
We are multiplying 2 times another number. This number
is the difference between some number and 8, written x – 8.
Using parentheses around this difference, the final expression
is 2(x – 8).
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1.2 – Slide 12
1.2 Variables, Expressions, and Equations
Translating Word Phrases to Algebraic Expressions
CAUTION
Notice that in translating the words “the difference between a
number and 8” in Example 3(f), the order is kept the same: x – 8.
“The difference between 8 and a number” would be written 8 – x.
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1.2 – Slide 13
1.2 Variables, Expressions, and Equations
Identifying Solutions of Equations
An equation is a statement that two expressions are equal.
An equation always includes the equality symbol, =.
x +4 = 11, 2y = 16, 4p + 1 = 25 – p
Equations
To solve an equation, we must find all values of the
variable that make the equation true.
Such values of the variable are called the solutions of
the equation.
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1.2 – Slide 14
1.2 Variables, Expressions, and Equations
Identifying Solutions of Equations
Example 4
Decide whether the given number is a solution of the equation.
(a) Is 7 a solution of 5p + 1 = 36?
5 p  1  36
5  7  1  36
35  1  36
36  36
Replace p with 7.
Multiply.
True
The number 7 is a solution of the equation.
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1.2 – Slide 15
1.2 Variables, Expressions, and Equations
Identifying Solutions of Equations
Example 4 (concluded)
Decide whether the given number is a solution of the equation.
(a) Is
14
3
a solution of 9m – 6 = 32?
9m  6  32
14
9   6  32
3
42  6  32
36  32
The number
14
3
Replace m with
14
.
3
Multiply.
False.
is not a solution of the equation.
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1.2 – Slide 16
1.2 Variables, Expressions, and Equations
Translating Sentences To Equations
Example 5
Write each word sentence as an equation. Use x as the variable.
(a) Twice the sum of a number and four is six.
“Twice” means two times. The word is suggests equals.
With x representing the number, translate as follows.
Twice
2·
the sum of a
number and four
is
six.
(x + 4)
=
6
2(x + 4) = 6
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1.2 – Slide 17
1.2 Variables, Expressions, and Equations
Translating Sentences To Equations
Example 5 (continued)
Write each word sentence as an equation. Use x as the variable.
(b) Nine more than five times a number is 49.
Use x to represent the unknown number. Start with 5x
and then add 9 to it. The word is translates as =.
5x + 9 = 49
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1.2 – Slide 18
1.2 Variables, Expressions, and Equations
Translating Sentences To Equations
Example 5 (concluded)
Write each word sentence as an equation. Use x as the variable.
(c) Seven less than three times a number is eleven.
Here, 7 is subtracted from three times a number to
get 11.
Three times
a number
less
seven
is
eleven.
3x
–
7
=
11
3x – 7 = 11
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1.2 – Slide 19
1.2 Variables, Expressions, and Equations
Distinguishing Between Expressions and Equations
Example 6
Decide whether each is an equation or an expression.
(a) 2x – 5y
There is no equals symbol, so this is an expression.
(b) 2x = 5y
Because there is an equals symbol with something on
either side of it, this is an equation.
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1.2 – Slide 20