Commutative Law
Algebraic Expressions and Algebraic Equations
An algebraic expression is a mathematical phrase that can contain ordinary numbers (constants), variables (like x or y) and operators, such as addition, subtraction, multiplication,
division or exponents.
Examples:
An algebraic
equation includes one or more variables and sets two algebraic expressions equal to each other.
Examples:
,
We will learn about simplifying
algebraic expressions and solving algebraic equations in Module 2.
Equivalent expressions
Consider the expressions
. Now let us
substitute x=3 in both expressions.
and
. Regardless of the value given for x, these two algebraic expressions will produce identical results.
Any two such algebraic expressions are said to be equivalent.
Commutative Law
The commutative law is valid only
for addition and multiplication and states that we can change the order of addition and mulitiplication, without changing the result. This law gives us the ability to re-arrange variables when solving equations. We will see how useful the commutative law is, later in this module.
Commutative
Law for Addition:
Since
both sums are equal to 12:
To generalize:
Commutative
Law for Multiplication :
Since
both products are equal to 20:
This
is true for all numbers.
To generalise:
This law can be applied to variables as well as constants. Now let us take a look at some examples:
Fill
in the blanks (Examples 1-5)
Example1:
Using the commutative law we know that the answer is 3 since:
Hence,
Example2:
Using the commutative law, we get
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Example 3:
Again using the commutative law we get
Example 4:
According to the commutative law
Example 5:
According to the commutative law
Example 6: Write an expression
equivalent to
The commutative law can be used in
a number of ways to write equivalent expressions:
Method 1: Using the commutative law for addition, we obtain :
Method 2: Using the commutative law for multiplication, we obtain:
Method 3: Using the commutative law for addition and multiplication, we obtain:
All these expressions are equivalent to
each other. That is,
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