Download Equivalent Expressions The Commutative and Associative Laws

Math 52
1.7 "Properties of Real Numbers"
Objectives:
*
Find equivalent fraction expressions and simplify fraction expressions.
*
Use the commutative and associative laws to …nd equivalent expressions.
*
Use the distributive laws to multiply and factor expressions.
*
Collect like terms.
Preliminaries:
In this section, we will consider several laws of real numbers that will allow us to …nd equivalent expressions.
The Identity Property of 0 :
For any real number a, a + 0 = 0 + a = a
The Identity Property of 1 :
For any real number a, a 1 = 1 a = a
Equivalent Expressions
De…nition:
"Equivalent Expressions"
kTwo expressions that have the same value for all allowable replacements are called equivalent.k
Example 1: (Equivalent expressions)
Write an equivalent fraction expression for.
a)
3
=
4
8
Example 2: (Simplify)
Simplify.
5xy
a)
40y
b)
b)
2
=
5
c)
15t
16m
12m
c)
2
=
3
3x
18p
24pq
The Commutative and Associative Laws
The Commutative Laws:
Addition:
For any numbers a and b;
Multiplication: For any numbers a and b;
Page: 1
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
1.7
Example 3: (Commutative law)
Use the commutative law to …nd an expression equivalent to:
a) xy + t
b) 9 + ab
The Associative Laws:
Addition:
For any numbers a; b and c;
Multiplication: For any numbers a; b and c;
Example 4: (Associative law)
Use the associative law to …nd an expression equivalent to:
a) (x + 5) + y
b) (6x) y
Example 5: (Associative and commutative laws)
Use the associative and commutative laws to write three expressions equivalent to:
a) y + (3 + x)
b) (xy) 3
The Distributive Laws
The distributive laws are the basis of many procedures in both arithmetic and algebra. They are probably the most
important laws that we use to manipulate algebraic expressions.
The Distributive Law of Multiplication Over Addition:
For any numbers a; b; and c;
The Distributive Law of Multiplication Over Subtraction:
For any numbers a; b; and c;
Example 6: (Using the distributive law)
Multiply.
a) 2 (x
2 + 3y)
b)
Page: 2
3
(p + q
5
5)
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
c)
7 ( 2x
5y + 9)
1.7
d) 3:1 ( 1:2x + 3:2y
Example 7: (Review)
Name the property or law illustrated by the equation.
Equation
a)
1:1)
Property
8x = x ( 8)
b) x + (4:3 + b) = (x + 4:3) + b
c) 0 + k = k
d) ( 8a) b =
8 (ab)
e) (p) (1) = p
f) 2 (t + 5) = 2t + 2 (5)
g) m + 34 = 34 + m
Factoring
Factoring
kTo factor an expression is to …nd an equivalent expression that is a product.k
Example 8: (Factor)
Factor the following algebraic expressions.
a) 32 4y
b) 9a + 27b
c) bx + by
d)
bz
81
3
1
x+ y
2
2
1
2
Collecting Like Terms
Example 9: (Combining like terms)
Combine or collect like terms.
a) x 0:24x
b) x + 4y
2x
c)
d)
5x + 6 + 3y
5x + 4y
2x
y
Page: 3
8x
2y
2y
4
Notes by Bibiana Lopez