mth 10905 algebra
DESCRIPTION
MTH 10905 Algebra. The Multiplication property of equality CHAPTER 2 Section 3. Identity Reciprocals. Reciprocal – two numbers are reciprocals when their product equals 1. If a is a non-zero number the reciprocal is - PowerPoint PPT PresentationTRANSCRIPT
MTH 10905Algebra
THE MULTIPLICATION PROPERTY OF EQUALITY
CHAPTER 2 SECTION 3
Identity Reciprocals
Reciprocal ndash two numbers are reciprocals when their product equals 1
If a is a non-zero number the reciprocal is
The reciprocal of a positive is positive and the reciprocal of a negative is a negative
1
a
Identity Reciprocals
The reciprocal of 0 does not exist
first we cannot have a zero on the bottom of a fraction
second zero divided by zero is zero
Exp the reciprocal of 3 is because
Exp the reciprocal of -2 is because
0 1 00
1 0 0
1
3
3 1 31
1 3 3
1
2
2 1 21
1 2 2
Identity Reciprocals
Exp the reciprocal of is because
Exp the reciprocal of is because
4 1 4 4 4 5 4 5 201 1 1
45 5 5 5 4 5 4 205
7
2
2
7
5
4
4
5
2 1 4 4 4 5 4 5 201 1 1
47 5 5 5 4 5 4 205
Multiplication Property to Solve Equation
Multiplication Property of Equalityif a = b then a c = b c for any real number a b and c
We can multiply any non-zero number to both sides without changing the solution
We can solve equations in the form of ax = b using the multiplication property
To isolate the variable we will multiply by the reciprocal of the numerical coefficient
Multiplication Property to Solve Equation
Exp Exp
1 12
3
3 1 312
1 3 1
3 1 12 3
1 3 1 1
36
x
x
x
x
3 4
12
4
1
1
12
1
4
4
1
124
x
x
x
x
Multiplication Property to Solve Equation
Exp Division is defined in the term of multiplication
this allows is to divide both sides by a non-zero
number
Exp
5 65
1 15 65
5 5
1 5 65 1
5 1 1 5
65
5
13
b
b
b
b
b
74
4 7 44
47 4
1 4
r 28
r
r
r
ba
b
a 1 means
Multiplication Property to Solve Equation
Exp Exp
3 9
4
4 3 49
3 4 3
4 3 9 4
3 4 1 3
36
3
12
x
x
x
x
x
9 5
1 19 ( ) 5
9 9
1 9 5 1
9 1 1 9
5
9
x
x
x
x
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Identity Reciprocals
Reciprocal ndash two numbers are reciprocals when their product equals 1
If a is a non-zero number the reciprocal is
The reciprocal of a positive is positive and the reciprocal of a negative is a negative
1
a
Identity Reciprocals
The reciprocal of 0 does not exist
first we cannot have a zero on the bottom of a fraction
second zero divided by zero is zero
Exp the reciprocal of 3 is because
Exp the reciprocal of -2 is because
0 1 00
1 0 0
1
3
3 1 31
1 3 3
1
2
2 1 21
1 2 2
Identity Reciprocals
Exp the reciprocal of is because
Exp the reciprocal of is because
4 1 4 4 4 5 4 5 201 1 1
45 5 5 5 4 5 4 205
7
2
2
7
5
4
4
5
2 1 4 4 4 5 4 5 201 1 1
47 5 5 5 4 5 4 205
Multiplication Property to Solve Equation
Multiplication Property of Equalityif a = b then a c = b c for any real number a b and c
We can multiply any non-zero number to both sides without changing the solution
We can solve equations in the form of ax = b using the multiplication property
To isolate the variable we will multiply by the reciprocal of the numerical coefficient
Multiplication Property to Solve Equation
Exp Exp
1 12
3
3 1 312
1 3 1
3 1 12 3
1 3 1 1
36
x
x
x
x
3 4
12
4
1
1
12
1
4
4
1
124
x
x
x
x
Multiplication Property to Solve Equation
Exp Division is defined in the term of multiplication
this allows is to divide both sides by a non-zero
number
Exp
5 65
1 15 65
5 5
1 5 65 1
5 1 1 5
65
5
13
b
b
b
b
b
74
4 7 44
47 4
1 4
r 28
r
r
r
ba
b
a 1 means
Multiplication Property to Solve Equation
Exp Exp
3 9
4
4 3 49
3 4 3
4 3 9 4
3 4 1 3
36
3
12
x
x
x
x
x
9 5
1 19 ( ) 5
9 9
1 9 5 1
9 1 1 9
5
9
x
x
x
x
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Identity Reciprocals
The reciprocal of 0 does not exist
first we cannot have a zero on the bottom of a fraction
second zero divided by zero is zero
Exp the reciprocal of 3 is because
Exp the reciprocal of -2 is because
0 1 00
1 0 0
1
3
3 1 31
1 3 3
1
2
2 1 21
1 2 2
Identity Reciprocals
Exp the reciprocal of is because
Exp the reciprocal of is because
4 1 4 4 4 5 4 5 201 1 1
45 5 5 5 4 5 4 205
7
2
2
7
5
4
4
5
2 1 4 4 4 5 4 5 201 1 1
47 5 5 5 4 5 4 205
Multiplication Property to Solve Equation
Multiplication Property of Equalityif a = b then a c = b c for any real number a b and c
We can multiply any non-zero number to both sides without changing the solution
We can solve equations in the form of ax = b using the multiplication property
To isolate the variable we will multiply by the reciprocal of the numerical coefficient
Multiplication Property to Solve Equation
Exp Exp
1 12
3
3 1 312
1 3 1
3 1 12 3
1 3 1 1
36
x
x
x
x
3 4
12
4
1
1
12
1
4
4
1
124
x
x
x
x
Multiplication Property to Solve Equation
Exp Division is defined in the term of multiplication
this allows is to divide both sides by a non-zero
number
Exp
5 65
1 15 65
5 5
1 5 65 1
5 1 1 5
65
5
13
b
b
b
b
b
74
4 7 44
47 4
1 4
r 28
r
r
r
ba
b
a 1 means
Multiplication Property to Solve Equation
Exp Exp
3 9
4
4 3 49
3 4 3
4 3 9 4
3 4 1 3
36
3
12
x
x
x
x
x
9 5
1 19 ( ) 5
9 9
1 9 5 1
9 1 1 9
5
9
x
x
x
x
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Identity Reciprocals
Exp the reciprocal of is because
Exp the reciprocal of is because
4 1 4 4 4 5 4 5 201 1 1
45 5 5 5 4 5 4 205
7
2
2
7
5
4
4
5
2 1 4 4 4 5 4 5 201 1 1
47 5 5 5 4 5 4 205
Multiplication Property to Solve Equation
Multiplication Property of Equalityif a = b then a c = b c for any real number a b and c
We can multiply any non-zero number to both sides without changing the solution
We can solve equations in the form of ax = b using the multiplication property
To isolate the variable we will multiply by the reciprocal of the numerical coefficient
Multiplication Property to Solve Equation
Exp Exp
1 12
3
3 1 312
1 3 1
3 1 12 3
1 3 1 1
36
x
x
x
x
3 4
12
4
1
1
12
1
4
4
1
124
x
x
x
x
Multiplication Property to Solve Equation
Exp Division is defined in the term of multiplication
this allows is to divide both sides by a non-zero
number
Exp
5 65
1 15 65
5 5
1 5 65 1
5 1 1 5
65
5
13
b
b
b
b
b
74
4 7 44
47 4
1 4
r 28
r
r
r
ba
b
a 1 means
Multiplication Property to Solve Equation
Exp Exp
3 9
4
4 3 49
3 4 3
4 3 9 4
3 4 1 3
36
3
12
x
x
x
x
x
9 5
1 19 ( ) 5
9 9
1 9 5 1
9 1 1 9
5
9
x
x
x
x
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Multiplication Property to Solve Equation
Multiplication Property of Equalityif a = b then a c = b c for any real number a b and c
We can multiply any non-zero number to both sides without changing the solution
We can solve equations in the form of ax = b using the multiplication property
To isolate the variable we will multiply by the reciprocal of the numerical coefficient
Multiplication Property to Solve Equation
Exp Exp
1 12
3
3 1 312
1 3 1
3 1 12 3
1 3 1 1
36
x
x
x
x
3 4
12
4
1
1
12
1
4
4
1
124
x
x
x
x
Multiplication Property to Solve Equation
Exp Division is defined in the term of multiplication
this allows is to divide both sides by a non-zero
number
Exp
5 65
1 15 65
5 5
1 5 65 1
5 1 1 5
65
5
13
b
b
b
b
b
74
4 7 44
47 4
1 4
r 28
r
r
r
ba
b
a 1 means
Multiplication Property to Solve Equation
Exp Exp
3 9
4
4 3 49
3 4 3
4 3 9 4
3 4 1 3
36
3
12
x
x
x
x
x
9 5
1 19 ( ) 5
9 9
1 9 5 1
9 1 1 9
5
9
x
x
x
x
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Multiplication Property to Solve Equation
Exp Exp
1 12
3
3 1 312
1 3 1
3 1 12 3
1 3 1 1
36
x
x
x
x
3 4
12
4
1
1
12
1
4
4
1
124
x
x
x
x
Multiplication Property to Solve Equation
Exp Division is defined in the term of multiplication
this allows is to divide both sides by a non-zero
number
Exp
5 65
1 15 65
5 5
1 5 65 1
5 1 1 5
65
5
13
b
b
b
b
b
74
4 7 44
47 4
1 4
r 28
r
r
r
ba
b
a 1 means
Multiplication Property to Solve Equation
Exp Exp
3 9
4
4 3 49
3 4 3
4 3 9 4
3 4 1 3
36
3
12
x
x
x
x
x
9 5
1 19 ( ) 5
9 9
1 9 5 1
9 1 1 9
5
9
x
x
x
x
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Multiplication Property to Solve Equation
Exp Division is defined in the term of multiplication
this allows is to divide both sides by a non-zero
number
Exp
5 65
1 15 65
5 5
1 5 65 1
5 1 1 5
65
5
13
b
b
b
b
b
74
4 7 44
47 4
1 4
r 28
r
r
r
ba
b
a 1 means
Multiplication Property to Solve Equation
Exp Exp
3 9
4
4 3 49
3 4 3
4 3 9 4
3 4 1 3
36
3
12
x
x
x
x
x
9 5
1 19 ( ) 5
9 9
1 9 5 1
9 1 1 9
5
9
x
x
x
x
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Multiplication Property to Solve Equation
Exp Exp
3 9
4
4 3 49
3 4 3
4 3 9 4
3 4 1 3
36
3
12
x
x
x
x
x
9 5
1 19 ( ) 5
9 9
1 9 5 1
9 1 1 9
5
9
x
x
x
x
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Multiplication Property to Solve Equation
Exp Exp
-24 6
1 6 1 24
6 1 6
1 24
6 1
-24
-6
4
w
w
w
w
w
025 150
1 150 1025 ( )
025 1 025
150
025
6
x
x
x
x
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b1for a fractions multiply both sides by the reciprocal
of a2for whole numbers divide both sides by a
Exp Exp
5 -4
8
4 5 1
4 8 4
5
32
x
x
x
2 -7
5
7 5 2 5( )
1 2 5 2
35
2
x
x
x
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Solve Equation in the form of ndashx = a
Remember that x = a is the same as 1x = aTherefore -x = a is the same as -1x = a
Exp Exp - 10
1 10
1 1
10
y
y
y
- 19
1 19
1 1
19
x
x
x
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
Do some steps Mentally to Solve Equations
As you become comfortable you can do some of the steps mentally
Exp Exp10 70
70
10
7
x
x
x
116
4
(16)(4)
64
x
x
x
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57
HOMEWORK 23
Page 118 ndash 119
9 11 19 25 31 35 49 57