# Solving Multi-Step Equations with Variables on Both Sides 04:00 minutes

**Video Transcript**

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Transcript
**Solving Multi-Step Equations with Variables on Both Sides**

Meet Emilio and Joanna. They're at a fish store. They've come to buy some scaly friends, and they each have the same amount of money.

Emilio makes his selection quickly. He buys two Rainbow Blinkeys and ten Grey Borings. Joanna loves Rainbow Blinkeys, so she buys five of them and just one Grey Boring. They both spend all of their money.

When Emilio sees Joanna’s purchase, he feels sad because he realizes he could have bought another Rainbow Blinkey. He wonders, to get one more Rainbow Blinkey, how many Grey Borings would he need to exchange?
To find out how many, Emilio can use a **multi-step equation with variables on both sides** of the equal sign.

### Multi-step Equation Example 1

Just to refresh your memory, an equation works like a scale. And to keep the scale **balanced**, **when you add or subtract one side of the scale, you must perform the same operation to the other side.**

Joanna bought five Rainbow Blinkeys and one Grey Boring. For the same amount of money, Emilio bought two Rainbow Blinkeys and ten Grey Borings so the scale is balanced.

Grey Borings cost $1 each. We can write the expression 5x + 1 = 2x + 10 to represent this situation, where the variable x represents the cost of a Rainbow Blinkey.

To find out how many Grey Borings are equal to the cost of one Rainbow Blinkey, you need to isolate the Xs. Using **inverse operations**, we can modify the equation so we will have all the Xs on one side of the equal sign and all the constants on the other side: 5x + 1 = 2x + 10.

- First, let’s subtract one from both sides of the equation: 5x = 2x + 9.
- Next, subtract 2x from both sides of the equation. The scale is still balanced: 3x = 9.
- To isolate the x, use the opposite of multiplication. Divide by 3 on both sides of the equation.

What is the result? 3x ÷ 3 = 9 ÷ 3x = 3, which means the cost of one Rainbow Blinkey is equal to the cost of three Grey Borings. Emilio can exchange three Grey Borings and get one more Rainbow Blinkey. Emilio is happy!

### Multi-step Equation Example 2

Now that we have reviewed the basics, let's take a look at a more complicated problem: 3 · (2x − 5) + 10 = 4x + 9.

- See the
**parentheses**? Use the**Distributive Property**and multiply each of the terms inside the parentheses by three: 2x · 3 = 6x and −5 · 3 = −15. So we get 6x − 15 + 10 = 4x + 9. - To stay organized, on each side of the equal sign, combine any like terms as you go through the problem. So we get: 6x minus 5 = 4x + 9.
- Next, use inverse operations to move the variables with
**coefficients**to one side of the equation and the**constants**to the other. Remember to be strategic when you choose which side to move the terms. - Alright, Let’s move! First we deal with the constant and add 5 to each side of the equal sign. So we get: 6x = 4x + 14. Now, subtract 4x from each side of the equal sign. 2x = 14.
- Next, use the inverse operation to solve for x. Divide both sides of the equation by 2, x = 7!

As you can see, it's important to go **step-by-step** when solving equations such as this. I wonder how the fish are doing. Let’s take a look.

So peaceful… Hold the fish food! What's going on? Wow, talk about fight or flight! Look at those fish move!

**All Video Lessons & Practice Problems in Topic**Equations »

1 comment-4y+2(y+5)=18