# 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.

$$\frac{7x}{7}=\frac{35}{7}$$

$$x=5$$

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.

$$\frac{16x}{16}=\frac{-48}{16}$$
$$x=-3$$

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

## Distributive Property Practice Questions

Question #1:

Solve the equation: $$3\left(2x+4\right)-5=4x-9$$.

$$x=8$$

$$x=-8$$

$$x=5$$

$$x=-5$$

Using the distributive property, distribute 3 on the left side of the equation.

$$3\left(2x+4\right)-5=4x-9$$
$$6x+12-5=4x-9$$

Now, combine like terms on the left side of the equation.

$$6x+12-5=4x-9$$
$$6x+7=4x-9$$

To get the variable terms on the left side, subtract 4x from both sides of the equation.

$$6x+7-4x=4x-9-4x$$
$$2x+7=-9$$

Now, solve the two-step equation for $$x$$. Subtract 7 from both sides.

$$2x+7-7=-9-7$$
$$2x=-16$$

Divide both sides of the equation by 2.

$$\frac{2x}{2}=\frac{-16}{2}$$
$$x=-8$$

Question #2:

Solve the equation: $$-3\left(2x+6\right)+4x=3\left(x+4\right)$$.

$$x=-6$$

$$x=6$$

$$x=2$$

$$x=-3$$

Using the distributive property, distribute –3 on the left side and 3 on the right side of the equation.

$$-3\left(2x+6\right)+4x=3\left(x+4\right)$$
$$-6x-18+4x=3x+12$$

Now, combine like terms on the left side of the equation.

$$-2x-18=3x+12$$

To get the variable terms on the left side, subtract 3x from both sides of the equation.

$$-2x-18-3x=3x+12-3x$$
$$-5x-18=12$$

Now, solve the two-step equation for $$x$$. Add 18 to both sides.

$$-5x-18+18=12+18$$
$$-5x=30$$

Divide both sides of the equation by –5.

$$\frac{-5x}{-5}=\frac{30}{-5}$$
$$x=-6$$

Question #3:

Solve the equation: $$-2\left(x+5\right)-3=2\left(3x+1\right)+2x$$.

$$x=-\frac{15}{8}$$

$$x=-\frac{8}{15}$$

$$x=\frac{2}{3}$$

$$x=-\frac{3}{2}$$

Using the distributive property, distribute –2 on the left side and 2 on the right side of the equation.

$$-2\left(x+5\right)-3=2\left(3x+1\right)+2x$$
$$-2x-10-3=6x+2+2x$$

Now, combine like terms on the left side and right side of the equation.

$$-2x-13=8x+2$$

To get the variable terms on the left side, subtract 8x from both sides of the equation.

$$-2x-13-8x=8x+2-8x$$
$$-10x-13=2$$

Now, solve the two-step equation for $$x$$. Add 13 to both sides.

$$-10x-13+13=2+13$$
$$-10x=15$$

Divide both sides of the equation by –10.

$$\frac{-10x}{-10}=\frac{15}{-10}$$
$$x=-\frac{15}{10}$$

$$x=-\frac{15\div5}{10\div5}=-\frac{3}{2}$$

Question #4:

Four more than the product of two and the sum of a number and 1 equals the product of three and the difference of the number and 2. If $$x$$ is the number, what is the value of $$x$$?

$$x=9$$

$$x=5$$

$$x=12$$

$$x=-12$$

The product of two and the sum of the number $$x$$ and 1 can be written by the algebraic expression $$2(x+1)$$. Four more than this expression is $$2\left(x+1\right)+4$$. The product of three and the difference of $$x$$ and 2 can be written by the expression $$3(x-2)$$. Since the 2 algebraic expressions are the same, we have the following equation.

$$2\left(x+1\right)+4=3(x-2)$$

To find the number, distribute 2 on the left side and 3 on the right side of the equation first.

$$2\left(x+1\right)+4=3\left(x-2\right)$$
$$2x+2+4=3x-6$$

Combine like terms on the left side of the equation.

$$2x+6=3x-6$$

To get the variable terms on the left side, subtract $$3x$$ from both sides of the equation.

$$2x+6-3x=3x-6-3x$$
$$-x+6=-6$$

Now, solve the two-step equation for x. Subtract 6 from both sides.

$$-x+6-6=-6-6$$
$$\frac{-x}{-1}=\frac{-12}{-1}$$
$$x=12$$

So, the value of the number is $$x=12$$.

Question #5:

The length of a rectangle is 3 more than the product of 2 and the sum of its width and 4. If the perimeter of the rectangle is 76 feet, what is its width? Hint: The perimeter, $$P$$, of a rectangle is $$P=2l+2w$$.

18 feet

9 feet

10 feet

6 feet

Let x be the width of the rectangle. Since the length of the rectangle is 3 more than the product of 2 and the sum of its width and 4, we can represent the length by the expression
$$2\left(x+4\right)+3$$. Now, substitute $$l=2\left(x+4\right)+3$$, $$w=x$$, and $$P=76$$ into the equation for the perimeter of a rectangle.

$$P=2l+2w$$
$$76=2[\left(2(x+4\right)+3]+2x$$

To find the width, distribute the 2 on the right side of the equation. Then, distribute 4.

$$76=2[\left(2(x+4\right)+3]+2x$$
$$76=4\left(x+4\right)+2\cdot3+2x$$
$$76=4x+16+6+2x$$

Combine like terms on the right side of the equation.

$$76=6x+22$$

Now, solve the two-step equation for $$x$$. Subtract 22 from both sides.

$$76-22=6x+22-22$$
$$54=6x$$

Divide both sides of the equation by 6.

$$\frac{54}{6}=\frac{6x}{6}$$
$$9=x$$

So, the width is 9 feet.