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Properties in Algebraic Expressions

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Today we're going to talk about the properties of algebraic expressions. We're going to start by reviewing the associative, commutative, and distributive properties of addition and multiplication and then we'll extend those properties to expressions involving variables and see how they work. So let's start by looking at the commutative property. In general, the commutative property tells us that it doesn't matter the order in which we add or multiply numbers together.

So for example, 3 plus 4 is the same as 4 plus 3. Both sides of these are equal to 7. Also works for multiplication. If I have a negative 2 times 6, that's the same as 6 times negative 2. Again, this is equal to negative 12. And this is equal to negative 12.

So we can extend this property to expressions involving variables. For example, if I have 5 plus x plus 3, using the commutative property, that should be the same as 5 plus 3 plus x. So if I combine my like terms here, 5 and 3, this will give me 8. And I can add my x. On this side, again, my like terms are still 5 and 3 so I can add them together and then bring down my plus x. So we can see that the commutative property also works for expressions involving variables.

Let's look at the associative property. In general, the associative property tells us that we can group or associate different terms in an expression and still have an equivalent expression. So for an addition expression, that might look something like 3 plus 4 plus 1. By the associative property, that should still be equal to 3 plus 4 plus 1. So 3 plus 4 will give me 7. If I add 1 to that, that gives me 8. And here, 4 plus 1 gives me 5. And if I add 3 to that, that gives me 8 also.

The associative property works for multiplication also. For example, 2 times 5 times 3 should be equal to 2 times 5 times 3. So 5 times 3 gives me 15. If I times that by 2, that gives me 30. And here 2 times 5 gives me 10. If I times that by 3, I also get 30. So it works for multiplication.

And we can see that it's also going to work for an expression with variables. For example, if I have 4y plus 6 grouped together and minus 2, by the associative property that should be equal to 4y plus 6 minus 2. So if I were to evaluate this, inside my parentheses I cannot actually combine the 6 and the 4y. So all I can do is combine the 6 and the minus 2, which would give me 4y. And then 6 minus 2 is 4. So plus 4.

On this side, I do have my like terms grouped together so 6 minus 2 gives me the 4. And then I can bring down my 4y and the plus. So again, we see that the associative property works for expressions involving variables.

So lastly, we're going to revisit the distributive property and the backward process from distributing, which is called factoring. So the distributive property tells us that if I have a number being multiplied by a sum of numbers, I can rewrite this by distributing my 8 to the 4 and the 8 to the 2 through multiplication. So this should be equivalent to 8 times 4 plus 8 times 2.

So we can see that this works. 4 plus 2 will give me 6. If I multiply that by 8, that gives me 48. And here, 8 times 4 will give me 32. 8 times 2 will give me 16. 32 plus 16 also gives me 48. So again, we can extend this distributive property to an expression with variables such as 5 times 2x plus 4. Using my distributive property, I can write this as 5 times 2x, which will give me 10x. And 5 times 4, which will give me 20.

So let's look at the backward process from distributing, which, again, we call factoring. And how factoring works is if I have two numbers, 12 plus 15, that have a common factor, a factor being a number that can go into or a number that you can multiply by another number to get the number that you're trying to factor out. So let's look at 12 and 15 have a common factor of 3, because 3 times 4 gives me 12 and 3 times 5 gives me 15.

So I'm going to factor out my 3 and then write the remaining factors on the inside of the parentheses. Again, 3 times a 4 will give me 12. And 3 times a 5 will give me 15. So you can see that this is, again, the backwards process from distributing. And so factoring will also work, again, for expressions involving variables. So if I have 6x plus 9, I'm going to look for a common factor, so something that goes into both 6 and 9 or both 6x and 9. And that number, again, is going to be 3. So if I factor out my 3, 3 times a 2x will give me 6x and 3 times another 3 will give me 9. So I can factor 6x plus 9 to be 3 times 2x plus 3.

So let's go over our key points from today. As always, make sure you get them in your notes so that you can refer to them later. So we saw that the associative, commutative, distributive properties, as well as factoring, can be extended to expressions that are involving variables. And we know that these properties will be useful when we're simplifying expressions and solving equations. So I hope that these notes and our examples helped you understand a little bit more about the properties of algebraic expressions. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.