Transcript Multiplication Property of Equality

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
1
a
 The reciprocal of a positive is positive and the reciprocal of a
negative is a negative
Identity Reciprocals
 The reciprocal of 0 does not exist.
 0  1  0
     0
 1  0  0
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
 Exp: the reciprocal of -2 is
1
3
because
1
2
 3  1  3
    1
 1  3  3
 2  1  2
because        1
 1  2  2
Identity Reciprocals
 Exp: the reciprocal of
4
5
is
5
4
because
 
 4   1   4  4   4  5   4  5  20
1
   4    1     1       
 5     5  5   5  4   5  4  20
 
5
 Exp: the reciprocal of 
2
7
is

7
2
because


 2   1   4  4   4  5   4  5  20
1
     4    1     1       
5   5 
4   5  4  20
 7 
 5 



 5
Multiplication Property to Solve Equation
 Multiplication Property of Equality
if 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:
4 x  12
 1  4 
 12  1 
   x    
 4  1 
 1  4 
12
x
4
x3
1
x  12
3
 3  1 
3
    x  12   
 1  3 
1
 3  1 
 12   3 
   x    
 1  3 
 1  1 
x  36
Multiplication Property to Solve Equation
 Exp:
5b  65
1
1
   5  b   65   
5
5
 1  5 
 65   1 
  b    
 5  1 
 1  5 
65
b
5
b  13
Division is defined in the term
of multiplication a means a  1
b
b
this allows is to divide both
sides by a non-zero number
r
Exp:
7
4
r
 4      7  4 
4
 4  r 
     7  4 
 1  4 
r  28
Multiplication Property to Solve Equation
 Exp:
3
x9
4
Exp:
9x  5
 4  3 
4
    x  9  
 3  4 
3
1
1
   9  ( x)   5   
9
9
 4  3 
 9  4 
   x    
 3  4 
 1  3 
 1  9 
 5  1 
    x     
 9  1 
 1  9 
36
x
3
x  12
5
x
9
Multiplication Property to Solve Equation
Exp:
-24  6 w
 1
 6  1 
    24    w      
 6
 1  6 
 1  24 
  
w
 6  1 
-24
w
-6
4w
Exp:
0.25 x  1.50
 1 
 1.50  1 

  0.25  ( x)  


 0.25 
 1  0.25 
x
1.50
0.25
x6
Multiplication Property to Solve Equation
When solving an equation in the form of ax = b:
1. for a fractions multiply both sides by the reciprocal of a
2. for whole numbers divide both sides by a
Exp:
5
-4x 
8
Exp:
2
-7   x
5
 4 x   5  1 

     
 4   8  4 
 7  5   2  5 
           ( x)
 1  2   5  2 
5
x
32
35
x
2
Solve Equation in the form of –x = a

Remember that x = a is the same as 1x = a
Therefore, -x = a is the same as -1x = a
Exp:
- y  10
1y 10

1 1
y  10
Exp:
- x  19
1x 19

1
1
x  19
Do some steps Mentally to Solve Equations

As you become comfortable you can do some of the steps
mentally
Exp:
10 x  70
70
x
10
x7
Exp:
1
x  16
4
x  (16)(4)
x  64
HOMEWORK 2.3
 Page 118 – 119
#9, 11, 19, 25, 31, 35, 49, 57