Transcript 3.1:Simplifying Algebraic Expressions

Chapter 3
Solving Equations
and Problem Solving
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
3.1
Simplifying
Algebraic
Expressions
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Constant and Variable Terms
A term that is only a number is called a constant term, or simply
a constant. A term that contains a variable is called a variable
term.
3y2 + (–4y) + 2
x+3
Constant
terms
Variable
terms
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Coefficients
The number factor of a variable term is called the numerical
coefficient. A numerical coefficient of 1 is usually not written.
5x
x or 1x
Numerical
coefficient is 5.
–7y
3y2
Numerical
coefficient is –7.
Understood
numerical
coefficient is 1.
Numerical
coefficient
is 3.
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Like Terms
Terms that are exactly the same, except that they may have
different numerical coefficients are called like terms.
Unlike Terms
5x, x 2
7x, 7y
5y, 5
6a, ab
Like Terms
3x, 2x
–6y, 2y, y
–3, 4
2ab2, –5b2a
The order of the variables
does not have to be the same.
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Distributive Property
A sum or difference of like terms can be simplified
using the distributive property.
Distributive Property
If a, b, and c are numbers, then
ac + bc = (a + b)c
Also,
ac – bc = (a – b)c
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Distributive Property
By the distributive property,
7x + 5x = (7 + 5)x
= 12x
This is an example of combining like terms.
An algebraic expression is simplified when all like
terms have been combined.
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Addition and Multiplication Properties
The commutative and associative properties of addition
and multiplication help simplify expressions.
Properties of Addition and Multiplication
If a, b, and c are numbers, then
Commutative Property of Addition
a+b=b+a
Commutative Property of Multiplication
a∙b=b∙a
The order of adding or multiplying two numbers can be
changed without changing their sum or product.
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Associative Properties
The grouping of numbers in addition or multiplication
can be changed without changing their sum or product.
Associative Property of Addition
(a + b) + c = a + (b + c)
Associative Property of Multiplication
(a ∙ b) ∙ c = a ∙ (b ∙ c)
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Helpful Hint
Examples of Commutative and Associative Properties
of Addition and Multiplication
4+3=3+4
Commutative Property of Addition
6∙9=9∙6
Commutative Property of Multiplication
(3 + 5) + 2 = 3 + (5 + 2) Associative Property of Addition
(7 ∙ 1) ∙ 8 = 7 ∙ (1 ∙ 8)
Associative Property of Multiplication
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Multiplying Expressions
We can also use the distributive property to multiply
expressions.
The distributive property says that multiplication
distributes over addition and subtraction.
2(5 + x) = 2 ∙ 5 + 2 ∙ x = 10 + 2x
or
2(5 – x) = 2 ∙ 5 – 2 ∙ x = 10 – 2x
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Simplifying Expressions
To simply expressions, use the distributive property
first to multiply and then combine any like terms.
Simplify: 3(5 + x) – 17
3(5 + x) – 17 = 3 ∙ 5 + 3 ∙ x + (–17)
Apply the Distributive
Property
= 15 + 3x + (–17)
Multiply
= 3x + (–2) or 3x – 2
Combine like terms
Note: 3 is not distributed to the –17 since –17
is not within the parentheses.
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Finding Perimeter
7z feet
3z feet
9z feet
Perimeter is the distance around the figure.
Perimeter = 3z + 7z + 9z
= 19z feet
Don’t forget to insert
proper units.
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Martin-Gay, Prealgebra, 6ed
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Finding Area
(2x – 5) meters
3 meters
A = length ∙ width
= 3(2x – 5)
= 6x – 15 square meters
Don’t forget to insert
proper units.
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Helpful Hint
Don’t forget . . .
Area:
• surface enclosed
• measured in square units
Perimeter:
• distance around
• measured in units
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