Algebra Topics

Simplifying Expressions

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#### Lesson 7: Simplifying Expressions

### Simplifying expressions

#### The order of operations

#### Adding like variables

#### The Distributive Property

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Simplifying an expression is just another way to say **solving a math problem**. When you **simplify** an expression, you're basically trying to write it in the **simplest** way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do. For example, take this expression:

4 + 6 + 5

If you **simplified** it by combining the terms until there was nothing left to do, the expression would look like this:

15

In other words, 15 is the **simplest** way to write 4 + 6 + 5. Both versions of the expression equal the exact same amount; one is just much shorter.

Simplifying **algebraic expressions** is the same idea, except you have variables (or letters) in your expression. Basically, you're turning a long expression into something you can easily make sense of. So an expression like this...

(13x + -3x) / 2

...could be simplified like this:

5x

If this seems like a big leap, don't worry! All you need to simplify most expressions is basic arithmetic -- addition, subtraction, multiplication, and division -- and the order of operations.

Like with any problem, you'll need to follow the** order of operations **when simplifying an algebraic expression. The order of operations is a rule that tells you the correct **order** for performing calculations. According to the order of operations, you should solve the problem in this order:

- Parentheses
- Exponents
- Multiplication and division
- Addition and subtraction

Let's look at a problem to see how this works.

In this equation, you'd start by simplifying the part of the expression in **parentheses**: 24 - 20.

2 ⋅ (24 - 20)^{2} + 18 / 6 - 30

**24** minus **20** is 4. According to the order of operations, next we'll simplify any **exponents**. There's one exponent in this equation: 4^{2}, or **four to the second power**.

2 ⋅ 4^{2} + 18 / 6 - 30

**4 ^{2} **is 16. Next, we need to take care of the

2 ⋅ 16 + 18 / 6 - 30

**2 ⋅ 16** is 32, and **18 / 6** is 3. All that's left is the last step in the order of operations: **addition** and **subtraction**.

32 + 3 - 30

**32 + 3** is 35, and **35 - 30** is 5. Our expression has been simplified—there's nothing left to do.

5

That's all it takes! Remember, you **must** follow the order of operations when you're performing calculations—otherwise, you may not get the correct answer.

Still a little confused or need more practice? We wrote an entire lesson on the order of operations. You can check it out here.

To add variables that are the same, you can simply **add the coefficients**. So** 3 x + 6x** is equal to 9

5y - 4y = 1y

You can also **multiply** and **divide** variables with coefficients. To multiply variables with coefficients, first multiply the coefficients, then write the variables next to each other. So **3 x ⋅ 4y** is 12

3x ⋅ 4y = 12xy

Sometimes when simplifying expressions, you might see something like this:

3(x+7)-5

Normally with the Order of Operations, we would simplify what is **inside** the parentheses first. In this case, however, x+7 can't be simplified since we can't add a variable and a number. So what's our first step?

As you might remember, the 3 on the outside of the parentheses means that we need to multiply everything **inside** the parentheses by 3. There are **two** things inside the parentheses: **x** and **7**. We'll need to multiply them **both** by 3.

3(x) + 3(7) - 5

3 · x is **3x** and 3 · 7 is **21**. We can rewrite the expression as:

3x + 21 - 5

Next, we can simplify the subtraction 21 - 5. 21 - 5 is **16**.

3x + 16

Since it's impossible to add variables and numbers, we can't simplify this expression any further. Our answer is **3x + 16**. In other words, 3(x+7) - 5 = 3x+16.

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