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Properties
2-1
Commutative Properties of Addition and Multiplication
Commutative Property of Addition
Commutative Property of Multiplication
For any number a and b,
For any numbers a and b,
The order in which numbers are added does
__________ change the sum.
The order in which numbers are multiplied
does ___________ change the product.
Associative Properties of Addition and Multiplication
Associative Property of Addition
Associative Property of Multiplication
For any numbers a, b, and c,
For any numbers a, b, and c,
The way in which addends are grouped does
not change the sum.
The way in which factors are grouped does
not change the product.
Identity Properties of Addition and Multiplication
For any number a,
For any number a,
The sum of an addend and zero, the additive
identity, is the addend.
The product of a factor and one, the
multiplicative identity, is the factor.
Multiplicative Property of Zero
For any number a,
The product of a factor and zero is zero.
2-1 continued
There have often been questions in math that you could do in your head and
others that require a great deal of work. Many times the properties are the reason
why some questions are easier to answer than others. We often will find
combinations of numbers that are easier to work with and then we either reorder or
recombine the numbers that we were given in a question.
Mental Math: Evaluate the expression. Justify each of your steps.
Example 1:
35 + 18 + 65 =
=
=
=
Example 2:
2,305,897  1 + 0 =
=
Using the properties, evaluate the expression when a = 7 and b = 25.
Example 3:
 2a  b   46 





Using the properties to simplify algebraic expressions
Example 4:
 4r  3 


Identify the property that the statement illustrates.
Example 5:
 2 6 3  3  2 6  _____________________ property of multiplication
2-1 continued
Multiplying by a conversion factor
Unit or dimensional analysis can be used to convert a given measurement to
different units by multiplying by conversion factors such as
Notice that each conversion factor equals one. Therefore, the value of the original
measurement does not change because of the identity property of multiplication.
Use a conversion factor to perform the indicated conversion
Example 6:
480 minutes to hours =
= ______ hours because all of the other labels canceled out.
The Distributive Property
2-2
Equivalent numerical expressions (71) –
Equivalent variable expressions (72) –
The Distributive Property states that the product of a sum or difference will be
equivalent to the sum or difference of the products.
a  b  c   ab  ac
 b  c  a  ba  ca
a(b  c)  ab  ac
(b  c)a  ba  ca
This property is the only one we will study that combines two operations… addition
and multiplication or subtraction and multiplication.
Use the distributive property to evaluate the expression.
Example 1
Example 2
5  7  3 
6 8  5 




Example 3
10  3 5  



Evaluate the expression using the distributive property and mental math.
Example 4
Example 5
6(89) =
(5.03)( 8 ) =
=
=
=
=
=
=
Use the distributive property to write an equivalent algebraic expression.
Example 6
Example 7
5  a  10  
5  w  8  




Example 8
3 a  b  
Example 9
 m  8 5 


Find the area of the rectangle or triangle.
Example 10
5
4m - 3
A  lw



Answer: The area is
(20m – 15) square units.
Example 11
1
A  bh
2
8
7 – 3k





Answer: The area is
_________square units.
Simplifying Variable Expressions
2-3
Term(s) (78) –
Coefficient (78) –
Constant term (78) –
Like terms (78) –
Simplest form –
Simplifying an expression –
For the given expression, identify the terms, like terms, coefficients, and constant
terms.
Example 1
3  7 x  2x  x
First rewrite the expression as a sum.
3  7 x  (2 x)  x
terms;
like terms;
coefficients;
constants;
Example 2
2a + 5c – a + 6a
First rewrite the expression as a sum.
2a + 5c +(– a) + 6a
terms;
like terms;
coefficients;
constants;
2-3 continued
Simplify each expression.
Example 3
8y  7  7 


Example 4
8  x  5x 




Example 5
2 x  3  3x  9 




Example 6
3 a  2   a 



Example 7
5c  (5c  2) 


Variables and Equations
2-4
Equation (85) –
Solution of an equation (85) –
Solving an equation (86) –
Just as we have previously practiced rewriting verbal statements as algebraic
expressions, we can also write them as equations.
Write the verbal sentence as an equation.
Example 1
The sum of x and 5 is 12.
Equation:
Example 2
The difference of 9 and a is 14.
Equation:
Example 3
The product of y and 12 is – 48.
Equation:
Example 4
The quotient of b and 3 is 15.
Equation:
Tell whether the given value of the variable is a solution of the equation.
Example 5
x  5  12; x
So, ____________________ a solution for the equation.
Example 6
9  a  14; a  5
So, ____________ a solution for the equation.
Solving Equations Using Addition and Subtraction
2-5
Inverse Operations (91) –
Equivalent Equations (91) –
Subtraction Property of Equality
If you subtract the same number from each side of an equation, the two sides
remain equal.
Solve each equation.
Example 1
Example 2
9  10  b
x  5  3
Example 3
6  t  4  9
Addition Property of Equality
If you add the same number to each side of an equation, the two sides remain
equal.
Example 4
Example 5
Example 6
r  5  10
19  g  18
3  x  20  63
Find the value of x.
Example 7
Perimeter = 50 ft.
16 ft.
x  16  25  50
25 ft.
x
So, x = _____ ft.
Solving Equations Using Multiplication or Division
2-6
Division Property of Equality
When you divide each side of an equation by the same nonzero number, the
two sides remain equal.
Example 1
Example 2
Example 3
4 y  32
56  7 p
4  2 w   112
Multiplication Property of Equality
When you multiply each side of an equation by the same number, the two
sides remain equal.
Example 4
f
 10
13
Example 5
k
 21 
8
Example 6
b
 11
8  11
Notice on examples 3 and 6, you must simplify before you solve.
Decimal Operations and Equations with Decimals
2-7
Some algebraic equations include decimal numbers. In order to solve such
equations, the rules you have previously learned still apply.
1. In order to add or subtract decimal numbers, the decimal points should be
___________________________.
2. In order to multiply decimal numbers, the product will have the same number
of decimal place values as the ______________ number of decimal place
values of the factors.
3. In order to divide decimal numbers, move the decimal point in the divisor to
the end and move the same number of movements in the ________________.
4. Use _________________ operations in order to solve any algebraic equation.
Example 1
3.6  (2.9)
Example 2
2.9  3.6
Example 5
8.1(0.3)
Example 6
8.1  (0.3) 
Example 7
m
 8.1
0.3
Example 3
m  3.6  2.9
Example 8
0.3m  8.1
Example 4
m  3.6  2.9