Solving Equations Using the Distributive Property

Hello! Welcome to this video about solving equations using the distributive property. As a quick refresher, remember, if you have something like this

\(3(5x+9)\)

 

the distributive property says that you can multiply this first term, the outer term, by every term inside the parentheses. So, if we did this we would get:

\(3\times 5x+3\times 9\)
 
\(15x+27\)

 

So, now that we’ve reviewed that, let’s use the distributive property to solve some equations. Let’s take a look at the equation:

\(2(3x-7)+x=21\)

 

So our first step is to simplify the distributive part. The 2 right here is going to be distributed to the \(3x\) and to the -7. Notice that it’s not distributed to this \(+x\) over here. That’s because it’s not inside the parentheses. You only multiply the number outside by each term inside the parentheses; the rest of the part of the equation stays the same.

\(6x-14+x=21\)

 

So now we can combine like terms on the left side. So \(6x\) and \(x\) are like terms.

\(7x-14=21\)

 

Now we can solve this just like a normal two-step equation. We’ll start by adding 14 to both sides.

\(7x-14+14=21+14\)
 
\(7x=35\)

 

And then all we have to do is divide by 7 on both sides.

\(\frac{7x}{7}=\frac{35}{7}\)
 
\(x=5\)

 

And that’s our answer!

Let’s look at another problem.

\(5(2x+6)=-2(3x+9)\)

 

For this one, we have to apply the distributive property to both sides of the equation, so we’re going to start by doing the left side.

\(10x+30=-2(3x+9)\)

 

Now, our equal sign stays the same and we’re going to apply the distributive property to the right side of the equation.

\(10x+30=-6x-18\)

 

Now we can solve it like a regular equation. So, we’re going to start by adding \(6x\) to both sides.

\(10x+6x+30=-6x+6x-18\)

 

Remember, we want to get all of our \(x\)-terms on one side, and all of our constants on the other side. However you do this is fine; if you wanted to subtract \(10x\) from both sides and get the \(x\)‘s on the right side instead of the left side, that’s totally fine, it would work as well. This is the way I’m going to work it out though; we’re going to get the \(x\)‘s on the left side and the constants on the right side.

\(16x+30=-18\)

 

Now, we’re going to subtract 30 from both sides, to move this over to the right side of the equation.

\(16x+30-30=-18-30\)
 
\(16x=-48\)

 

And finally, we divide by 16 on both sides.

\(\frac{16x}{16}=\frac{-48}{16}\)
\(x=-3\)

 

And that’s our answer!

I hope that this video was helpful. Thanks for watching and happy studying!

 

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by Mometrix Test Preparation | This Page Last Updated: July 11, 2022