Solving Equations and Problem Solving

1
Solving Equations and Problem Solving
Chapter Three
2
Simplifying Algebraic Expressions
Section 3.1
3
In algebra letters called variables represent
numbers.

• The addends of an algebraic expression are called
  the terms of the expression.

x 3
3y2 (- 4y) 2
Martin-Gay, Prealgebra, 5ed
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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
Martin-Gay, Prealgebra, 5ed
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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 - 7y 3y 2
Numerical coefficient is 5.
Numerical coefficient is -7.
Understood numerical coefficient is 1.
Numerical coefficient is 3.
Martin-Gay, Prealgebra, 5ed
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Terms that are exactly the same, except that they
may have different numerical coefficients are
called like terms.
Unlike Terms
Like Terms
3x, 2x - 6y, 2y, y - 3, 4
5x, x 2
7x, 7y 5y, 5 6a, ab
2ab2, - 5b 2a
The order of the variables does not have to be
the same.
Martin-Gay, Prealgebra, 5ed
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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

Martin-Gay, Prealgebra, 5ed
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• 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.

Martin-Gay, Prealgebra, 5ed
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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.

Martin-Gay, Prealgebra, 5ed
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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)

Martin-Gay, Prealgebra, 5ed
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Examples of Commutative and Associative
Properties of Addition and Multiplication
4 3 3 4 6 ? 9 9 ? 6 (3 5) 2 3 (5
2) (7 ? 1) ? 8 7 ? (1 ? 8)
Commutative property of Addition
Commutative property of Multiplication
Associative property of Addition
Associative property of Multiplication
Martin-Gay, Prealgebra, 5ed
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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

Martin-Gay, Prealgebra, 5ed
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To simply expressions, use the distributive
property first to multiply and then combine any
like terms.
Simplify 3(5 x) - 17
Apply the distributive property

• 3(5 x) - 17

3 ? 5 3 ? x (- 17)
15 3x (- 17)
Multiply
3x (- 2) or 3x - 2
Combine like terms
Martin-Gay, Prealgebra, 5ed
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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
Dont forget to insert proper units.
Martin-Gay, Prealgebra, 5ed
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Finding Area
A length ? width
3(2x 5) 6x 15
square meters
Dont forget to insert proper units.
Martin-Gay, Prealgebra, 5ed
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Dont forget . . .

• Area
• surface enclosed
• measured in square units

• Perimeter
• distance around
• measured in units