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Algebra I. Lesson 1
Definitions and Properties (Grades 9-12)

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Instruction 1-3

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Variables and Expressions | Open Sentences | Distributive and Additive Identity, Multiplicative Identity Property and Properties of Equality | Commutative and Associative Property | Summary

DISTRIBUTIVE AND ADDITIVE IDENTITY, MULTIPLICATIVE IDENTITY PROPERTY AND  PROPERTIES OF EQUALITY
CA GR7 AF 1.3, CA HS Algebra 1.1, 4.0,  25.2
In this and the next sections, we will discuss the properties of real numbers. These properties are the core of the all algebraic operations. Almost in any algebraic concepts, some of these properties can be observed. Knowing the logic of these properties helps the student to understand all the operations in a reasoning way.
 
Let look at some samples. In all the expressions below, the results of the both sides in each equality are the same.  
4(5 + 11) = 4(5) + 4(11)
 
12(3 + 21) = 12(3) + 12(21)
 
31(11 + 23) = 31(11) + 31(23)
 
If you review the forms of operations in both sides of each equality, you will find a pattern for all equalities. We define this pattern as below for all real numbers. 
 
Adding any number to another number produces a new number if neither of these numbers are zero. But adding 0 to any number does not make any change in the number. For this reason, 0 is called the additive identity. We generalized the property of the additive identity as below.



 
Let review the following examples.
4 × algl1_3_files/i0020000.jpg = 1
7 × algl1_3_files/i0020001.jpg = 1
11 × algl1_3_files/i0020002.jpg = 1
129 × algl1_3_files/i0020003.jpg = 1
In each example, the product of a number and its reciprocal is 1.
 
 Now, look at the following examples.
 
            4 × 1 = 4
 
            7 × 1 = 7
 
            11 × 1 = 11
 
            129 × 1 = 129
 
In all cases, the product of each number and 1 is the same as the number itself. In other words, multiplying any number by 1 does not change it. So, 1 is called multiplicative identity. Its property is summarized below.


Equalities have some common properties that are used in most algebraic operations. These properties are so important that without them algebra would not be developed.
 
Below, we describe all these properties. Understanding the logic of these properties will help you to learn the other topics easily.



 
 Example 1. Let m and n be real numbers. Then
 
a.      if m = a ,and m = 4  then a = 4
 
b.      if b = 9, then 9 = b
 
c.      if  c = d, then d = c


 


 


Example 2. If m + 3n = 4 and 4 = 3p + q, then m + 3n = 3p + q.

 

 

Example 3. Let x = 3 and y(x + 12) = 56. Then y(3 + 12) = 56 or 15y = 56.
 
Practice 4. Identify the property used in each row of the table below.
 

EQUALITY
PROPERTY
If m + 4n = 5x + 3y, then
5x + 3y = m + 4n
 
12x + 4m = 12x + 4m
 
If a + m = 12p and 12p = 4s + t,
then a + m = 4s + t
 
If x = 3 and 2x + 3y = 34,
then 2(3) + 3y = 34
  

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for Students, Parents and Teachers

Now let's do Practice Exercise 1-3 (top).

Next Page: Commutative and Associative Property (top)