Algebra Topics: Simplifying Expressions

Lesson 7: Simplifying Expressions


Simplifying expressions

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:


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:


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.

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:

  1. Parentheses
  2. Exponents
  3. Multiplication and division
  4. 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: 42, or four to the second power.

2 ⋅ 42 + 18 / 6 - 30

42 is 16. Next, we need to take care of the multiplication and division. We'll do those from left to right: 2 ⋅ 16 and 18 / 6.

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.


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.

Adding like variables

To add variables that are the same, you can simply add the coefficients. So 3x + 6x is equal to 9x. Subtraction works the same way, so 5y - 4y = 1y, or just y.

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 3x ⋅ 4y is 12xy.

3x ⋅ 4y = 12xy

The Distributive Property

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


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.