Simplify variable expressions involving like terms and the distributive property

Example: 2(5x+4)-3x

Example (Click to try)

2(5x+4)-3x

How to simplify your expression

To simplify your expression using the Simplify Calculator, type in your expression like 2(5x+4)-3x.
The simplify calculator will then show you the steps to help you learn how to simplify your algebraic expression on your own.

Typing Exponents

Type ^ for exponents like x^2 for "x squared". Here is an example:
2x^2+x(4x+3)

Simplifying Expressions Video Lesson

• Khan Academy Video: Simplifying Expressions

Need more problem types? Try MathPapa Algebra Calculator

How Do You Multiply Binomials Using the Distributive Property?

Multiplying together two binomials? Not a fan of the FOIL method, or just want to see another way? Check out this tutorial! You'll see how to distribute one binomial into the other in order to find the product. You get the same answer no matter which method you use, so be sure to add this method to your arsenal!

The Order of Operations and Variables:

Even without knowing what a variable is, we can sometimes make expressions with variables look simpler. This is done by simplifying our expression.

Here is a vocabulary word that will help you understand the lesson better:

• Coefficient = the number being multiplied to a variable (in 2n, 2 is the coefficient)
• Reduce = combine or simplify by doing whatever operations we can
• Term = a part of an expression separated from the rest by addition (in 3a + 6b, 3a is one term and 6b is another term)
• Like Terms = any terms in an expression where the variables are the same (3a and 4a, \(2{\text{b}}^{2}\) and \(5{\text{b}}^{2}\), note that \(2{\text{b}}^{2}\) and 3b are not like terms)

Video Source (09:10 mins) | Transcript

Remember to follow the order of operations. Sometimes this means to use the distributive property to solve what’s in the parentheses.

When we see two different letters, we can easily know that we don’t have like terms, but can we add \(3{\text{a}} + 4{\text{a}}^{2}\) ? Let’s say \({\text{a}}=3\), then \({\text{a}}^{2}=9\). Because these are different numbers the answer is no, we cannot add \(3{\text{a}}+4{\text{a}}^{2}\). Any time we have different letters as our variables, or the same letter with different powers, we do not have like terms.

Additional Resources

• Khan Academy: Intro to Combining Like Terms (04:32 mins, Transcript)
• Khan Academy: Simplifying Expressions (04:06 mins, Transcript)
• Khan Academy: Combining Like Terms - Challenge Problem (04:38 mins, Transcript)

Practice Problems

Simplify the following expressions:

1. 7w − 2w


2. 5s − 7 − 3s + 11


3. 5a − 2b − 6 + 3a + 6b


4. \(2{\text{v}}^{2}+6+3{\text{v}}{-}3{\text{v}}^{2}\)


5. \( 2(3-2{\text{t}}) + 5 ({\text{t}} + 3) \)


6. \( ( 4 {\text{x}} + 3 {\text{y}} - 2{\text{z}} ) - 2 ( {\text{x}} + 3 {\text{z}}) \)

Learning Outcomes

• Apply the distributive property to simplify an algebraic expression involving whole numbers, integers, fractions and decimals
• Apply the distributive property in different forms

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need [latex]$9.25[/latex]; that is, [latex]9[/latex] dollars and [latex]1[/latex] quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

If you think about doing the math in this way, you are using the Distributive Property.

Distributive Property

If [latex]a,b,c[/latex] are real numbers, then

[latex]a\left(b+c\right)=ab+ac[/latex]

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

[latex]3(9.25)\\3(9\quad+\quad0.25)\\3(9)\quad+\quad3(0.25)\\27\quad+\quad0.75\\27.75[/latex]

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression [latex]3\left(x+4\right)[/latex], the order of operations says to work in the parentheses first. But we cannot add [latex]x[/latex] and [latex]4[/latex], since they are not like terms. So we use the Distributive Property, as shown in the next example.

example

Simplify: [latex]3\left(x+4\right)[/latex]

Solution:

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the previous example would look like this:

[latex]3\cdot x+3\cdot 4[/latex]

Now you try.

try it

In our next example, there is a coefficient on the variable y. When you use the distributive property, you multiply the two numbers together, just like simplifying any product. You will also see another example where the expression in parentheses is subtraction, rather than addition.  You will need to be careful to change the sign of your product.

example

Simplify: [latex]6\left(5y+1\right)[/latex]

Simplify: [latex]2\left(x - 3\right)[/latex]

Now you try.

try it

The distributive property can be used to simplify expressions that look slightly different from [latex]a\left(b+c\right)[/latex]. Here are two other forms.

different Forms of the Distributive Property

If [latex]a,b,c[/latex] are real numbers, then

[latex]a\left(b+c\right)=ab+ac[/latex]

Other forms

[latex]a\left(b-c\right)=ab-ac[/latex]
[latex]\left(b+c\right)a=ba+ca[/latex]

In the following video we show more examples of using the distributive property.

Using the Distributive Property With Fractions and Decimals

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples. The distributive property comes in all shapes and sizes, and can include fractions or decimals as well.

example

Simplify: [latex]\Large\frac{3}{4}\normalsize\left(n+12\right)[/latex]

Simplify: [latex]8\Large\left(\frac{3}{8}\normalsize x+\Large\frac{1}{4}\right)[/latex].

Now you try.

try it

Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

example

Simplify: [latex]100\left(0.3+0.25q\right)[/latex]

Now you try.

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Distributing a Variable

In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

example

Simplify: [latex]m\left(n - 4\right)[/latex]

Now you try.

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The Backwards Form of the Distributive Property

The next example will use the ‘backwards’ form of the Distributive Property, [latex]\left(b+c\right)a=ba+ca[/latex].

example

Simplify: [latex]\left(x+8\right)p[/latex]

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Distributing a Negative Term

When you distribute a negative number, you need to be extra careful to get the signs correct.

example

Simplify: [latex]-2\left(4y+1\right)[/latex]

Simplify: [latex]-11\left(4 - 3a\right)[/latex]

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In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, [latex]-a=-1\cdot a[/latex].

example

Simplify: [latex]-\left(y+5\right)[/latex]

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How do you simplify the distributive property with variables?

What are like terms in distributive property?