Solving Equations Involving the Distributive Property

In this lesson, we will look at an example of how to solve and check an equation involving the distributive property.

 

Remember — Distributive Property

tells us that we can remove the parentheses by distributing (by multiplying) the number in front of parentheses to each term inside the parentheses.

For example one way to write the formula for finding
the perimeter of a rectangle is:             $\color{purple}{P=2(l+w)}$
or we could distribute the 2 and get:  $\color{purple}{P=2l+2x}$
 
Area of a Rectangle picture
 

Try these two problems:

1. Dan had 61 ft fencing that he wants to use to surround his garden. If the width is x ft. and the length is 1.5x+8 ft. what are the dimensions of his garden?
 
2. $5x-3(2x-4)=21$
 

Solutions:

1. Original problem:   Dan had $61$ ft fencing that he wants to use to surround his garden. If the width is $x$ ft. and the length is $1.5x+8$ ft. what are the dimensions of his garden?

a. Start by sketching out a picture of what is happening in the word problem: garden-sketch
b. Perimeter of a rectangle formula: $ \color{purple}P=2\color{blue}l+2\color{green}w$
c. Substitute the values into the formula: $ \color{purple}{61}=2\color{blue}{(1.5x+8)}+2\color{green}x$
d. Remove the parentheses by distributing the 2: $ 61=\color{#ff1493}2\color{blue}{(1.5x+8)}+2x$
$ 61=\color{#ff1493}{3x+16}+2x$
e. Combine like terms: $ 61=\color{green}{5x}+16$
f. Solve for $x$: $ 61\color{red}{-16}=5x+16\color{red}{-16} $
$ 45=5x$
$ 9=x$
So the garden is:   $13.5$ ft. by $9$ ft.

 
 

2.  Original problem $ 5x \color{red}{-}3(2x\color{red}{-} 4)=21$
a. Change all the subtraction problems to adding the opposite: $5x \color{blue}{+{^-3}}(2x\color{blue}{+{^-4}})=21$
b. Remove the parentheses by distributing the $^-3$ $5x +\color{#ff1493}{{^-6x}+12}=21$
c. Combine like terms: $\color{green}{^-1x}+12=21$
d. Solve for $x$ ${^-1x}+12\color{purple}{-12}=21\color{purple}{-12}$
${^-1x}=9$
$\color{red}{x={^-9}}$