How to Simplify an Expression: A Beginner's Guide | Algebraic Expressions | Math with Mr. J

Welcome to math with Mr. J.

In this video, I'm going to cover how to simplify algebraic expressions.

I'll cover how to combine like terms and how to use the distributive property in order to do so.

We will start with an introduction to combining like terms.

Then we will take a look at more examples.

After that, we will take a look at an introduction to the distributive property.

Following that introduction, we will take a look at more examples.

And then lastly, we will simplify expressions by using both combining like terms and the distributive property.

Now remember, like terms are terms with the same variables to the same powers.

When we combine like terms, we look for any like terms in the given algebraic expression and combine them into one term.

By combining like terms, we can simplify expressions.

That just means we can rewrite the original expression in a simpler and easier way to understand and work with.

Let's jump into number one where we have 9x + 3x.

We will start with this basic expression and work our way up.

So we have two terms in this expression, 9x and 3x.

Both terms have the same variable of x, and these variables of x are to the same power.

Remember, when we don't have an exponent attached to a variable, there is an understood exponent of one.

Anything to the power of one is just itself.

So 9x and 3x are like terms.

Now when we combine like terms, all we need to do is add or subtract the coefficients.

The numbers in front of the variables.

The coefficients in number one are 9 and 3.

We have a positive 9x plus a positive 3x, so let's add those coefficients.

9 + 3 is 12, and then we have the variable of x.

And that's it. We took those two like terms, 9x and 3x, and combined them into one term.

12x.

12x is equivalent to 9x + 3x, so we didn't change the value of the expression.

So 12x is our final simplified expression.

Let's move on to number two where we have 8g + 7 + 5g + 2.

Are there any like terms that we can combine in order to simplify this expression?

Yes.

We have 8g and 5g. Both of those terms have that variable of g,

and then we have constant terms, 7 and 2.

I'll box in the constant terms to separate them from the 8g and the 5g.

Now we can combine like terms.

We have 8g + 5g, that gives us 13g,

and then we have 7 + 2.

That gives us 9.

So we end up with 13g + 9, and that's our simplified expression.

That expression of 13g + 9 is equivalent to the original expression.

We were just able to simplify the original expression by combining like terms.

We started with four total terms, but we were able to combine like terms,

and now we only have two total terms.

Let's move on to number three where we have 6y² + 10y + 2y² + 3y + y.

Let's find any like terms that we can combine.

We'll start with 6y².

2y² is a like term.

Both of those terms have that variable of y to the power of 2.

Now, do we have any other like terms within this algebraic expression that we can combine?

Yes,

10y, and I will box these terms in in order to separate them from the y² terms.

3y, and then y.

Now I do want to mention this term right here, the y, the variable by itself,

the coefficient is 1.

We don't have a coefficient written in front.

Whenever you see that, the coefficient is 1.

And it can be helpful to write that one in there when you combine like terms.

So you can always write that one if you would like.

Now since this algebraic expression has five terms and we are working our way up to more complicated algebraic expressions,

we're going to use a strategy to help us organize the expression before we combine like terms.

We are going to rearrange and rewrite the expression and put the like terms next to each other.

I'll start with 6y² + the like term of 2y²

Now we have the y terms, so 10y + 3y + 1y.

So now all of the like terms are next to each other and it's a little easier to see what we can combine.

So this is a strategy to keep in mind.

Now, do you have to do this step in order to combine like terms?

No, but it can be helpful.

Now we can combine like terms.

We will start with 6y² + 2y².

So add the coefficients.

6 + 2 is 8, and then we have y².

Now we can combine the y terms.

So we have 10 + 3 + 1.

10 + 3 is 13, + 1 is 14.

So we get 8y² + 14y.

And that's the simplified expression.

We now have an equivalent expression that is simpler than the original.

We simplified the expression. We went from five terms to two terms.

Let's move on to number four where we have 7x + 2y - 4x + 2y.

Let's find any like terms that we can combine.

We will start with 7x and -4x.

Now when we combine like terms, a term is going to take the sign that's in front of it.

So this is -4x.

Then we have 2y and 2y.

So let's box those terms in in order to separate them from the x terms.

Now we can rewrite this expression with the like terms next to each other.

We will start with 7x - 4x + 2y + another 2y.

Now we can combine like terms.

We have 7x - 4x, or you can think of this as 7x being combined with -4x.

However you want to think about it.

7 - 4 is 3, and then we have the x.

Or if you're thinking about it as 7x combined with a -4x, 7 and -4 give us 3 as well.

Then we have our 2y + 2y, that gives us + 4y.

So we end up with 3x + 4y, and that's our simplified expression.

We went from four total terms to two total terms by combining like terms.

3x + 4y is equivalent to the original expression.

We were just able to again, simplify this expression by combining like terms.

Now I also want to go through simplifying this expression a slightly different way to start off.

And that's by rewriting the original expression with only addition separating the terms.

We do this by changing any subtraction to adding the opposite.

The benefit of having all terms separated only by addition is that it's a little simpler to identify all of the terms,

especially any negative terms.

It kind of organizes the expression and helps any negatives stand out.

I'll rewrite the expression off to the side here.

So 7x + 2y - 4x + 2y.

So let's rewrite subtraction as adding the opposite.

So adding the opposite of a positive 4x is a negative 4x.

So adding the opposite.

Let's rewrite the expression with that change.

So we have 7x + 2y + -4x + 2y.

Now we can combine like terms.

We have 7x + -4x, that gives us 3x.

And then we have 2y + 2y, so that gives us + 4y.

3x + 4y that way as well.

So that's just another strategy to be aware of.

So there's an introduction to combining like terms.

Let's move on to the distributive property.

Here is an introduction to the distributive property.

Now the distributive property can help us remove parentheses within algebraic expressions.

This helps us simplify expressions when we do not have like terms within parentheses that we can combine.

The distributive property works when we have addition or subtraction inside of the parentheses.

So at the top of the screen, there is a general overview of the distributive property where a is being distributed to the terms inside of the parentheses.

The distributive property and that overview will make a lot more sense as we go through our examples.

Let's jump into number one where we have two and then in parentheses 5 + 3.

And we're going to do this two different ways.

By using the order of operations, so doing what's in the parentheses first,

and then also using the distributive property.

Now for number one, we don't have any variables involved.

We are actually able to add what's in the parentheses first and then go from there.

We don't have to use the distributive property.

But the point of number one is to show us that we get the same thing either way.

This is going to show us that the distributive property doesn't change the value of an expression.

We are able to use this strategy.

So again, we get the same thing either way.

Let's start by using the order of operations and doing what's in the parentheses first.

We have 5 + 3, which is 8.

Bring down the two.

And now we have 2 * 8, which is 16.

Now let's use the distributive property and see if we still get 16.

So we need to take that two on the outside of the parentheses and distribute it to the 5 and to the 3.

So we have 2 * 5 + 2 * 3.

2 * 5 gives us 10.

+ 2 * 3 gives us 6.

10 + 6 is 16.

So we get 16 that way as well.

So we can see that the distributive property doesn't change the value of an expression,

and we are able to use it.

Let's move on to number two where we have 8 and then in parentheses 2m + 6.

Now we can't combine those terms in the parentheses.

So what we can do, we can use the distributive property to remove those parentheses and simplify this expression.

So let's distribute the 8 to the 2m and to the 6.

This gives us 8 * 2m + 8 * 6.

8 * 2m is 16m.

+ 8 * 6 is 48.

Now 16m and 48 are unlike terms.

So we don't have any terms that we can combine.

So we are done here. 16m + 48 is our simplified expression.

Let's move on to number three where we have 7 and then in parentheses a - 9.

Let's distribute that 7 to the a and to the 9.

That gives us 7 * a - 7 * 9.

7 * a is just 7a.

- 7 * 9 is 63.

So we end up with 7a - 63 that way as well.

And again, that's just a different way to think through it.

You get the same thing either way, but you can include the sign in front of the term and think of that as a -9.

So something to keep in mind.

Let's move on to number four where we have 10 and then in parentheses -5x - 4y.

Let's distribute the 10 to the -5x and to the 4y.

So 10 * -5x - 10 * 4y.

10 * -5x gives us -50x.

So we end up with -50x - 40y.

Now let's take a look at a different way to think through this.

So I will rewrite the expression off to the side here.

We need to distribute the 10 to the -5x and then we will think of that as -4y.

So include the sign in front of that term.

10 * -5x is -50x.

And then 10 * -4y is -40y.

So we get the same thing that way as well.

-50x - 40y.

So there is an introduction to the distributive property.

Let's take a look at four more examples.

Here are four more algebraic expressions that we need to simplify using both the distributive property and combining like terms.

These will get a little more complex than the previous four examples.

Let's jump into number one where we have 13a + 4 and then in parentheses a + 9.

Now since we have parentheses, we need to start there.

We can't combine the terms in the parentheses, they are unlike terms.

So we can use the distributive property to remove the parentheses.

Once the parentheses are removed, we can look to combine like terms.

So let's distribute that 4 to the a and to the 9.

So we have 4 * a, which is 4a.

And then 4 * 9 is 36.

So + 36.

And then we can bring down 13a.

Now that the parentheses are removed, we can look to combine like terms in order to simplify this further.

So, do we have any like terms that we can combine?

Yes.

13a and 4a are like terms.

So we can combine those terms.

13a + 4a gives us 17a.

And then we have + 36.

And this is our final simplified expression.

17a + 36.

Now that simplified expression is equivalent to the original expression.

We were just able to simplify that original expression by using the distributive property and combining like terms.

Let's move on to number two where we have 5 and then in parentheses x² - 3.

And then + 10 - 4x.

Let's start by using the distributive property in order to remove the parentheses.

We're going to distribute the 5 to the x² and to the -3.

5 * x² gives us 5x².

And then 5 * -3 gives us -15.

Now another way to think through that distributive property there is to do 5 * x² which is 5x².

Bring the subtraction sign down, and then do 5 * 3.

We get 5x² - 15 that way as well.

Then we have + 10 - 4x.

Now that we removed the parentheses, we can look to combine like terms.

So, do we have any like terms that we can combine?

Yes, we have two constant terms, -15 and 10.

So let's combine those like terms.

-15 + 10 or -15 combined with positive 10.

That gives us -5.

So -5.

And then we have 5x².

And then -4x.

And this is our final simplified expression.

5x² - 4x - 5.

Now I do want to mention as far as how this simplified expression is written.

Typically speaking, when writing expressions, the greatest exponent comes first.

So greatest to least.

If exponents are the same, go in ABC order.

Constant terms go last.

So that's something to keep in mind as far as writing out expressions.

Let's move on to number three where we have 7 and then in parentheses g + 3h.

+ 4 and then in parentheses 2g - 6h.

Let's start by using the distributive property to remove any parentheses.

We're going to distribute the 7 to the g and to the 3h.

7 * g gives us 7g.

And then 7 * 3h gives us 21h.

So + 21h.

Then we can distribute the 4 to the 2g and to the -6h.

4 * 2g gives us 8g.

So + 8g.

And then 4 * -6h gives us -24h.

So -24h.

Now all of the like terms are right next to each other.

So like I mentioned, it's a little simpler to combine the like terms.

So now we can combine like terms.

Let's start with 7g and 8g.

7g + 8g gives us 15g.

Then we have 21h - 24h.

21h - 24h gives us -3h.

And this is our final simplified expression.

15g - 3h.

Lastly, let's move on to number four where we have 18x - 10 and then in parentheses 2x - 2y + 9.

And then -6x.

Let's start by using the distributive property to remove the parentheses.

We're going to distribute -10 to the 2x, to the -2y, and to the 9.

-10 * 2x gives us -20x.

-10 * -2y gives us a positive 20y.

Remember, a negative times a negative equals a positive.

And then -10 * 9 gives us -90.

So -90.

We then have the -6x that we need to bring down.

Now we can look for any like terms that we can combine.

So we have 18x, -20x, and -6x.

Those are like terms.

We can combine those.

18x - 20x - 6x gives us -8x.

Then we have a positive 20y.

We don't have any other like terms to combine with 20y.

So we bring that down.

And then we have -90.

We don't have any other like terms to combine with -90.

So we bring that down.

So our final simplified expression is -8x + 20y - 90.

So there's how to simplify algebraic expressions by combining like terms and using the distributive property.

I hope that helped.

Thanks so much for watching.

Until next time.

Peace.