Associative Law

The associative law is valid only for addition and multiplication and states that we can change the grouping of numbers being added or mulitiplied, without changing the result. This law gives us the ability to re-arrange variables when solving equations. We will see how useful the associative law is, later in this module.

Associative Law for Addition:

$\begin{array}{l}\left(5+7\right)+3=12+3=15\text{ (grouping 5 and 7 first and then adding 3)}\\ 5+(7+3)=5+10=15\text{ (adding 3 to 7 first and then adding 5)}\end{array}$

Since both sums are equal to 15:

$\left(5+7\right)+3=5+(7+3)$

This is true for all numbers.

Generalizing:$a+(b+c)=(a+b)+c\text{ for any real numbers a,b and c}$

Associative Law for Multiplication:

$\begin{array}{l}(5\times 4)\times 2=20\times 2=40\text{ (multiply 4 with 5 first, then mutiply the result with 2)}\\ 5\times (4\times 2)=5\times 8=40\text{ (multiply 4 with 2 first, then mutiply the result with 5)}\end{array}$

Since both products are equal to 40:

$(5\times 4)\times 2=5\times (4\times 2)$

This is true for all numbers.

Generalizing:$(a\times b)\times c=a\times (b\times c)\text{ for any real numbers a,b and c}$

This law can be applied to variables as well as constants. Let's take a look at some examples.

Fill in the blanks (Examples 1-5)

Example1: $(3\times 4)\times 2=3\times (\_\_\times 2)$

Solution:

Example 2: $(4\times 3)\times y=4\times (3\times \_\_)$

Solution:

Example 3: $$$(2+4)+5=\_\_+(4+5)$

Solution:

Example 4: $(a+3)+7=\_\_+(3+7)$

Solution:

Example 5: Use the commutative and associative laws to write at least three expressions equivalent to $3+(a+b)$

Solution: