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Simplifying algebraic expressions


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

We will show you first how to simplify numerical expressions.

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

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

If you had the following problem to add 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

What does this have to do with simplifying algebraic expressions?

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

Well, in the same manner, just count how many v's 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 v's, we call the v's like terms. Same for the x's, 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.

You are ready to do some challenging problems

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 xy's 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 ot change anything

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

Any questions about simplifying algebraic expressions? Contact me.





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