Simplifying algebraic expressions

Simplifying algebraic expressions with one or more variables by combining like terms is what this lesson will show you.  

Learning first how to simplify numerical expressions will make it easier to understand how to simplify algebraic expressions.

Let's say you want to simplify 5 + 5 + 5 + 5. What is a quick way to get the answer?

You can do 4 × 5 and 4 × 5 is a simplified version of 5 + 5 + 5 + 5.

If you had the following problem to simplify 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5, you can quickly see why it would be useful to just count how many fives there are and then times the amount by 5.

Since there are 11 fives, the simplified expression is 11 × 5

Relationship between simplifying numerical expressions and simplifying algebraic expressions

Try to simplify v + v + v + v + v + v

Well, in the same manner, just count how many vs there are and multiply the amount by v.

Since there are 6 of them, v + v + v + v + v + v = 6 × v.

To simplify 6 × v even further, you can write 6v. However, remember that there is a multiplication between 6 and v.

Notice that putting a 1 next to each v will not change the answer.

v + v + v + v + v + v = 1v + 1v + 1v + 1v + 1v + 1v = ( 1 + 1 + 1 + 1 + 1 + 1)v = 6v

This shows you a couple of things. First, 1v = v. Then, when simplifying algebraic expressions, you can just do your math with the number on the left on the variable.

Now, try to add v + v + v + v + x + x + x

I hope it makes sense to you that you cannot add v and x. What is v + x? You cannot say 2v because it is v + v that is equal to 2v.

You cannot say 2x because it is x + x that is equal to 2x. Basically v and x are not like terms, so v + x is just equal to v + x.

By the same token, there is another thing you need to watch for. What is v + 2? It is not 2v as already shown.

Again, v and 2 are not like terms either, so v + 2 = v + 2

However, since you can add all the vs, we call the vs like terms. Same for the xs, they are called like terms.

v + v + v + v + x + x + x = ( v + v + v + v) + ( x + x + x) = 4 × v + 3 × x = 4v + 3x

What about 4v + 3x + 5v + 4x?

You cannot add 4v to 3x, but you can add 4v to 5v. And you can add 3x to 4x

4v + 3x + 5v + 4x = 4v + 5v + 3x + 4x = 9v + 7x

When simplifying algebraic expressions, subtraction and/or putting negative numbers work pretty much the same. You can only subtract like terms. The only difficulty you will encounter is the subtraction or addition of integers.

Do 4v - 3x - 2v + 4x

I can do 4v with 2v. However, be very careful! The operation on the left of 2v is subtraction, so you have to do 4v - 2v = 2v

In the same way, I can do 3x with 4x, but the operation next to 3x is subtraction. You have to change the subtraction to + -

Then, you will do -3x + 4x = 1x. I hope you have mastered how to add integers

4v - 3x - 2v + 4x = 2v + 1x

Useful guidelines when simplifying algebraic expressions.

Step 1: Look for like terms

Step 2: Watch out for minus or negative sign next to variables (on the left of the variables)

Step 3: When you move this term around, you got to move it with the minus or negative sign as well

Step 4: Add or subtract only the integers on the left of the like terms.

More challenging problems about simplifying algebraic expressions.


Example #1

9x + 4 + 15v - 7x + 8 - 10v

Put like terms together

9x + 4 + 15v - 7x + 8 - 10v = 9x - 7x + 15v - 10v + 4 + 8 = 2x + 5v + 12

Example #2

-5 - 6x - 5x + 10y - 4x + 1 + 10x + y

Put like terms together

-5 - 6x - 5x + 10y - 4x + 1 + 10x + y = -6x - 5x - 4x + 10x + 10y + y - 5 + 1

Just add all the numbers next to x, called coefficients.

-6 - 5 - 4 + 10 = -11 - 4 + 10 = -15 + 10 = -5

Then, add all the numbers next to y.

10 + 1 = 11

-5 - 6x - 5x + 10y - 4x + 1 + 10x + y = -5x + 11y + -4

More on like terms when simplifying algebraic expressions

When simplifying algebraic expressions, as long as you are adding the same thing, you have like terms even if you see more than one variable in each term.

For example what is xy + xy + xy + xy + xy + xy?

How many xys do you see? Since there are 6 of them, xy + xy + xy + xy + xy + xy = 6xy

We do have a multiplication between x and y, but it does not change anything.

In the same way, 8xyzf + 12xyzf + xyzf + xyzf + xyzf = 20xyzf + 3xyzf = 23xyzf

Simplifying algebraic expressions quiz. 

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