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Section 1.4
Basic Rules
of Algebra
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Objective 1
Understand and use the vocabulary of
algebraic expressions.
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Vocabulary of Algebraic Expressions
• Terms: The terms of an algebraic expression
are those parts that are separated by addition.
– In the expression 2 x  8, 2 x is a term and 8 is
a term.
• Coefficient: The numerical part of a term.
– In the term 2x, 2 is the coefficient.
• Like terms: Like terms have exactly the same
variable part.
– 5x and 8x are like terms.
– 2y and 7y are like terms.
– 2x and 3y are not like terms.
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Objective 1: Example
1.
Use the algebraic expression to answer the
following questions.
6 x  2 x  11
a. How many terms are in the algebraic
expression?
There are 3 terms.
b. What is the coefficient of the first term?
6 is the coefficient of the first term.
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Objective 1: Example
Use the algebraic expression to answer the
following questions.
6x  2x  11
c. What is the constant term?
11 is the constant term.
d. What are the like terms in the
algebraic expression?
6x and 2x are like terms.
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Objective 2
Use commutative properties.
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Commutative Property
• Let a and b represent real numbers, variables or
algebraic expressions.
– Commutative Property of Addition
ab ba
Changing order in addition does not affect the
sum.
– Commutative Property of Multiplication
ab  ba
Changing order in multiplication does not
affect the product.
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Objective 2: Example
2a.
Use the commutative property of addition to
write an equivalent algebraic expression of
x  14.
14  x
2b.
Use the commutative property of
multiplication to write an equivalent
algebraic expression of 7y.
y7
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Objective 3
Use associative properties.
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Associative Property
• Let a, b, and c represent real numbers,
variables or algebraic expressions.
– Associative Property of Addition
(a  b )  c  a  ( b  c )
Changing grouping when adding does not
affect the sum.
– Associative Property of Multiplication
(ab )c  a(bc )
Changing grouping when multiplying does
not affect the product.
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Objective 3: Example
3a.
Simplify: 6(5x)
6(5 x )  (6  5) x
 30x
3b.
Simplify: 8  ( x  4)
8  ( x  4)  8  (4  x )
 (8  4)  x
 12  x or x  12
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Objective 4
Use the distributive property.
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Distributive Property
• Let a, b, and c represent real numbers,
variables or algebraic expressions.
a(b  c )  ab  ac
Multiplication distributes over addition.
• Example:
7( x  3)  7 x  7  3  7 x  21
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Objective 4: Example
4a.
Multiply: 5( x  3)
5( x  3)  5  x  5  3
 5x  15
4b.
Multiply: 6(4 y  7)
6(4 y  7)  6  4 y  6  7
 24 y  42
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Objective 5
Combine like terms.
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Combining Like Terms
Add or subtract the coefficients of the terms.
Example:
7x  3x
The terms are like terms, therefore to combine the
terms, add the coefficients, 7  3  10. This is the
coefficient of the combined term.
7x  3x  10x
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Objective 5: Example
5a.
Combine like terms: 7x  3x
7 x  3 x  (7  3) x
 10 x
5b.
Combine like terms: 9a  4a
9a  4a  (9  4)a
 5a
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Example
Simplify: 2 x  8  5 x  7
 (2 x  5 x )  (8  7)
 7 x  15
Rearrange terms and
group like terms using
the commutative and
associative properties.
Combine like terms.
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Example
Simplify: 3 y  2 x  4 x  7 y
 (2 x  4 x )  (3 y  7 y ) Rearrange terms and
group like terms using
the commutative and
associative properties.
 6 x  10 y
Combine like terms.
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Objective 5: Example
5c.
Simplify: 9 x  6 y  5 x  2y
9 x  6 y  5 x  2y  (9 x  5 x )  (6 y  2y )
 14 x  8 y
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Objective 6
Simplify algebraic expressions.
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Algebraic Expressions
Simplifying Algebraic Expressions
1. Use the distributive property to remove
parentheses.
2. Rearrange terms and group like terms using
the commutative and associative properties.
This step may be done mentally.
3. Combine like terms by combining the
coefficients of the terms and keeping the same
variable factor.
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Example
Simplify: 3(2 x  5)  7
 6 x  15  7
 6x  8
Use the distributive property
to remove the parentheses.
Combine like terms.
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Example
Simplify: 3(2a  4b )  2(2a  3b)
Use the distributive
 6a  12b  4a  6b
property to remove
the parentheses.
Use the associative
 (6a  4a )  (12b  6b )
and commutative
properties to
rearrange and group
like terms.
Combine like terms.
 10a  18b
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Example
Simplify: 3a  (2a  4b  6c)  6b  3c
 3a  2a  4b  6c  6b  3c
 (3a  2a )  (6b  4b )  (6c  3c )
 3  2 a   6  4  b   6  3  c
 1a  2b  3c
 a  2b  3c
Distributive Prop.
Comm. &
Assoc. Prop
Distributive Prop.
Subtract
Simplify
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Objective 6: Example
6a. Simplify : 7(2 x  3)  11x
7(2 x  3)  11x  7  2 x  7  3  11x
 14 x  21  11x
 (14 x  11x )  21
 25 x  21
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Objective 6: Example
6b. Simplify : 7(4 x  3 y )  2(5 x  y )
7(4 x  3 y )  2(5 x  y )  7  4 x  7  3 y  2  5 x  2  y
 28 x  21y  10 x  2y
 (28 x  10 x )  (21y  2y )
 38 x  23 y
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