Which property justifies each step? (6+2)a + 8b = 8a + 8b.

Question:

Which property justifies each step? {eq}(6+2)a + 8b = 8a + 8b. {/eq}

Distributive Property

An algebraic expression is an expression which consists of constants, variables and algebraic operations like addition, subtraction, product and division.

Algebraic expressions follow distributive law.

If there are two linear algebraic expressions as

{eq}\displaystyle f(x)\ =\ ax\ +\ b\\ \displaystyle g(x)\ =\ cx\ +\ d {/eq}

then according to the distributive property, their product can be written as

{eq}\displaystyle f(x)\ \times\ g(x)\ =\ (ax\ +\ b)\ \times\ ( cx\ +\ d)\ =\ ax\ \times\ (cx\ +\ d)\ +\ b\ \times\ (cx\ +\ d) {/eq}

Commutative Property

According to the Commutative Property, if any operation is operated between two numbers, let's say a and b then the result will be same if it is applied between b and a.

Mathematically, according to this property,

{eq}\displaystyle a\ f\ b\ =\ b\ f\ a {/eq}

where:

• f is the operation like addition or multiplication

Associative Property

We can multiply or add three or more than three numbers without considering the positions of the numbers.

For example, if we have three numbers as, a, b and c then using this property we can write

{eq}\displaystyle a\ +\ (b\ +\ c)\ =\ (a\ +\ b)\ +\ c\ =\ (a\ +\ c)\ +\ b\\ \displaystyle a\ \times\ (b\ \times\ c)\ =\ (a\ \times\ b)\ \times\ c\ =\ (a\ \times\ c)\ \times\ b {/eq}

Here we have,

{eq}(6+2)a + 8b = 8a + 8b. {/eq}

The first term shows the distribution property.