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In this video, I'm going to cover how to simplify algebraic expressions.
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I'll cover how to combine like terms and how to use the distributive property in order to do so.
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We will start with an introduction to combining like terms.
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Then we will take a look at more examples.
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After that, we will take a look at an introduction to the distributive property.
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Following that introduction, we will take a look at more examples.
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And then lastly, we will simplify expressions by using both combining like terms and the distributive property.
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Now remember, like terms are terms with the same variables to the same powers.
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When we combine like terms, we look for any like terms in the given algebraic expression and combine them into one term.
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By combining like terms, we can simplify expressions.
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That just means we can rewrite the original expression in a simpler and easier way to understand and work with.
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Let's jump into number one where we have 9x + 3x.
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We will start with this basic expression and work our way up.
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So we have two terms in this expression, 9x and 3x.
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Both terms have the same variable of x, and these variables of x are to the same power.
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Remember, when we don't have an exponent attached to a variable, there is an understood exponent of one.
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Anything to the power of one is just itself.
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So 9x and 3x are like terms.
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Now when we combine like terms, all we need to do is add or subtract the coefficients.
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The numbers in front of the variables.
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The coefficients in number one are 9 and 3.
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We have a positive 9x plus a positive 3x, so let's add those coefficients.
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9 + 3 is 12, and then we have the variable of x.
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And that's it. We took those two like terms, 9x and 3x, and combined them into one term.
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12x.
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12x is equivalent to 9x + 3x, so we didn't change the value of the expression.
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So 12x is our final simplified expression.
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Let's move on to number two where we have 8g + 7 + 5g + 2.
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Are there any like terms that we can combine in order to simplify this expression?
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Yes.
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We have 8g and 5g. Both of those terms have that variable of g,
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and then we have constant terms, 7 and 2.
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I'll box in the constant terms to separate them from the 8g and the 5g.
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Now we can combine like terms.
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We have 8g + 5g, that gives us 13g,
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and then we have 7 + 2.
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That gives us 9.
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So we end up with 13g + 9, and that's our simplified expression.
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That expression of 13g + 9 is equivalent to the original expression.
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We were just able to simplify the original expression by combining like terms.
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We started with four total terms, but we were able to combine like terms,
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and now we only have two total terms.
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Let's move on to number three where we have 6y² + 10y + 2y² + 3y + y.
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Let's find any like terms that we can combine.
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We'll start with 6y².
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2y² is a like term.
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Both of those terms have that variable of y to the power of 2.
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Now, do we have any other like terms within this algebraic expression that we can combine?
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Yes,
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10y, and I will box these terms in in order to separate them from the y² terms.
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3y, and then y.
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Now I do want to mention this term right here, the y, the variable by itself,
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the coefficient is 1.
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We don't have a coefficient written in front.
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Whenever you see that, the coefficient is 1.
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And it can be helpful to write that one in there when you combine like terms.
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So you can always write that one if you would like.
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Now since this algebraic expression has five terms and we are working our way up to more complicated algebraic expressions,
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we're going to use a strategy to help us organize the expression before we combine like terms.
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We are going to rearrange and rewrite the expression and put the like terms next to each other.
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I'll start with 6y² + the like term of 2y²
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Now we have the y terms, so 10y + 3y + 1y.
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So now all of the like terms are next to each other and it's a little easier to see what we can combine.
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So this is a strategy to keep in mind.
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Now, do you have to do this step in order to combine like terms?
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No, but it can be helpful.
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Now we can combine like terms.
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We will start with 6y² + 2y².
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So add the coefficients.
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6 + 2 is 8, and then we have y².
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Now we can combine the y terms.
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So we have 10 + 3 + 1.
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10 + 3 is 13, + 1 is 14.
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So we get 8y² + 14y.
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And that's the simplified expression.
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We now have an equivalent expression that is simpler than the original.
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We simplified the expression. We went from five terms to two terms.
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Let's move on to number four where we have 7x + 2y - 4x + 2y.
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Let's find any like terms that we can combine.
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We will start with 7x and -4x.
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Now when we combine like terms, a term is going to take the sign that's in front of it.
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So this is -4x.
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Then we have 2y and 2y.
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So let's box those terms in in order to separate them from the x terms.
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Now we can rewrite this expression with the like terms next to each other.
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We will start with 7x - 4x + 2y + another 2y.
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Now we can combine like terms.
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We have 7x - 4x, or you can think of this as 7x being combined with -4x.
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However you want to think about it.
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7 - 4 is 3, and then we have the x.
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Or if you're thinking about it as 7x combined with a -4x, 7 and -4 give us 3 as well.
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Then we have our 2y + 2y, that gives us + 4y.
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So we end up with 3x + 4y, and that's our simplified expression.
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We went from four total terms to two total terms by combining like terms.
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3x + 4y is equivalent to the original expression.
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We were just able to again, simplify this expression by combining like terms.
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Now I also want to go through simplifying this expression a slightly different way to start off.
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And that's by rewriting the original expression with only addition separating the terms.
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We do this by changing any subtraction to adding the opposite.
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The benefit of having all terms separated only by addition is that it's a little simpler to identify all of the terms,
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especially any negative terms.
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It kind of organizes the expression and helps any negatives stand out.
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I'll rewrite the expression off to the side here.
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So 7x + 2y - 4x + 2y.
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So let's rewrite subtraction as adding the opposite.
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So adding the opposite of a positive 4x is a negative 4x.
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So adding the opposite.
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Let's rewrite the expression with that change.
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So we have 7x + 2y + -4x + 2y.
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Now we can combine like terms.
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We have 7x + -4x, that gives us 3x.
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And then we have 2y + 2y, so that gives us + 4y.
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3x + 4y that way as well.
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So that's just another strategy to be aware of.
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So there's an introduction to combining like terms.
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Let's move on to the distributive property.
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Here is an introduction to the distributive property.
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Now the distributive property can help us remove parentheses within algebraic expressions.
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This helps us simplify expressions when we do not have like terms within parentheses that we can combine.
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The distributive property works when we have addition or subtraction inside of the parentheses.
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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.
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The distributive property and that overview will make a lot more sense as we go through our examples.
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Let's jump into number one where we have two and then in parentheses 5 + 3.
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And we're going to do this two different ways.
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By using the order of operations, so doing what's in the parentheses first,
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and then also using the distributive property.
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Now for number one, we don't have any variables involved.
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We are actually able to add what's in the parentheses first and then go from there.
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We don't have to use the distributive property.
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But the point of number one is to show us that we get the same thing either way.
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This is going to show us that the distributive property doesn't change the value of an expression.
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We are able to use this strategy.
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So again, we get the same thing either way.
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Let's start by using the order of operations and doing what's in the parentheses first.
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We have 5 + 3, which is 8.
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Bring down the two.
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And now we have 2 * 8, which is 16.
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Now let's use the distributive property and see if we still get 16.
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So we need to take that two on the outside of the parentheses and distribute it to the 5 and to the 3.
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So we have 2 * 5 + 2 * 3.
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2 * 5 gives us 10.
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+ 2 * 3 gives us 6.
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10 + 6 is 16.
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So we get 16 that way as well.
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So we can see that the distributive property doesn't change the value of an expression,
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and we are able to use it.
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Let's move on to number two where we have 8 and then in parentheses 2m + 6.
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Now we can't combine those terms in the parentheses.
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So what we can do, we can use the distributive property to remove those parentheses and simplify this expression.
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So let's distribute the 8 to the 2m and to the 6.
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This gives us 8 * 2m + 8 * 6.
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8 * 2m is 16m.
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+ 8 * 6 is 48.
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Now 16m and 48 are unlike terms.
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So we don't have any terms that we can combine.
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So we are done here. 16m + 48 is our simplified expression.
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Let's move on to number three where we have 7 and then in parentheses a - 9.
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Let's distribute that 7 to the a and to the 9.
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That gives us 7 * a - 7 * 9.
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7 * a is just 7a.
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- 7 * 9 is 63.
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So we end up with 7a - 63 that way as well.
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And again, that's just a different way to think through it.
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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.
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So something to keep in mind.
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Let's move on to number four where we have 10 and then in parentheses -5x - 4y.
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Let's distribute the 10 to the -5x and to the 4y.
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So 10 * -5x - 10 * 4y.
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10 * -5x gives us -50x.
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So we end up with -50x - 40y.
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Now let's take a look at a different way to think through this.
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So I will rewrite the expression off to the side here.
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We need to distribute the 10 to the -5x and then we will think of that as -4y.
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So include the sign in front of that term.
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10 * -5x is -50x.
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And then 10 * -4y is -40y.
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So we get the same thing that way as well.
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-50x - 40y.
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So there is an introduction to the distributive property.
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Let's take a look at four more examples.
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Here are four more algebraic expressions that we need to simplify using both the distributive property and combining like terms.
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These will get a little more complex than the previous four examples.
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Let's jump into number one where we have 13a + 4 and then in parentheses a + 9.
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Now since we have parentheses, we need to start there.
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We can't combine the terms in the parentheses, they are unlike terms.
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So we can use the distributive property to remove the parentheses.
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Once the parentheses are removed, we can look to combine like terms.
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So let's distribute that 4 to the a and to the 9.
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So we have 4 * a, which is 4a.
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And then 4 * 9 is 36.
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So + 36.
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And then we can bring down 13a.
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Now that the parentheses are removed, we can look to combine like terms in order to simplify this further.
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So, do we have any like terms that we can combine?
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Yes.
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13a and 4a are like terms.
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So we can combine those terms.
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13a + 4a gives us 17a.
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And then we have + 36.
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And this is our final simplified expression.
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17a + 36.
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Now that simplified expression is equivalent to the original expression.
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We were just able to simplify that original expression by using the distributive property and combining like terms.
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Let's move on to number two where we have 5 and then in parentheses x² - 3.
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And then + 10 - 4x.
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Let's start by using the distributive property in order to remove the parentheses.
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We're going to distribute the 5 to the x² and to the -3.
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5 * x² gives us 5x².
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And then 5 * -3 gives us -15.
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Now another way to think through that distributive property there is to do 5 * x² which is 5x².
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Bring the subtraction sign down, and then do 5 * 3.
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We get 5x² - 15 that way as well.
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Then we have + 10 - 4x.
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Now that we removed the parentheses, we can look to combine like terms.
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So, do we have any like terms that we can combine?
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Yes, we have two constant terms, -15 and 10.
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So let's combine those like terms.
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-15 + 10 or -15 combined with positive 10.
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That gives us -5.
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So -5.
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And then we have 5x².
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And then -4x.
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And this is our final simplified expression.
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5x² - 4x - 5.
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Now I do want to mention as far as how this simplified expression is written.
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Typically speaking, when writing expressions, the greatest exponent comes first.
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So greatest to least.
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If exponents are the same, go in ABC order.
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Constant terms go last.
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So that's something to keep in mind as far as writing out expressions.
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Let's move on to number three where we have 7 and then in parentheses g + 3h.
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+ 4 and then in parentheses 2g - 6h.
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Let's start by using the distributive property to remove any parentheses.
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We're going to distribute the 7 to the g and to the 3h.
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7 * g gives us 7g.
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And then 7 * 3h gives us 21h.
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So + 21h.
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Then we can distribute the 4 to the 2g and to the -6h.
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4 * 2g gives us 8g.
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So + 8g.
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And then 4 * -6h gives us -24h.
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So -24h.
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Now all of the like terms are right next to each other.
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So like I mentioned, it's a little simpler to combine the like terms.
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So now we can combine like terms.
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Let's start with 7g and 8g.
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7g + 8g gives us 15g.
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Then we have 21h - 24h.
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21h - 24h gives us -3h.
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And this is our final simplified expression.
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15g - 3h.
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Lastly, let's move on to number four where we have 18x - 10 and then in parentheses 2x - 2y + 9.
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And then -6x.
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Let's start by using the distributive property to remove the parentheses.
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We're going to distribute -10 to the 2x, to the -2y, and to the 9.
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-10 * 2x gives us -20x.
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-10 * -2y gives us a positive 20y.
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Remember, a negative times a negative equals a positive.
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And then -10 * 9 gives us -90.
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So -90.
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We then have the -6x that we need to bring down.
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Now we can look for any like terms that we can combine.
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So we have 18x, -20x, and -6x.
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Those are like terms.
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We can combine those.
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18x - 20x - 6x gives us -8x.
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Then we have a positive 20y.
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We don't have any other like terms to combine with 20y.
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So we bring that down.
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And then we have -90.
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We don't have any other like terms to combine with -90.
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So we bring that down.
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So our final simplified expression is -8x + 20y - 90.
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So there's how to simplify algebraic expressions by combining like terms and using the distributive property.
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I hope that helped.
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Thanks so much for watching.
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Until next time.
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Peace.
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