Main content

### Course: Operations and Algebraic Thinking 222-226 > Unit 2

Lesson 2: The distributive property & equivalent expressions- The distributive property with variables
- Factoring with the distributive property
- Distributive property with variables (negative numbers)
- Create equivalent expressions by factoring
- Equivalent expressions: negative numbers & distribution
- Equivalent expressions: negative numbers & distribution
- Equivalent expressions

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Equivalent expressions: negative numbers & distribution

Learn how to identify equivalent expressions using your knowledge of the distributive property and negative numbers.

## Want to join the conversation?

- This has been bothering me for ages, and a thanks to Sal Khan for answering my other 24(maybe it was 240, I'm not sure)...but I still have one question...

1. I'm not sure if this appeals to THIS video... But it's related. Sometimes Sal changes a negative into a minus sign... How is that possible?

Any andhelp will be appreciated,**all**

If you answered this, Thanks!!!(42 votes)- It would have been a + -12, so instead he just made it -12, as they are the exact same.(8 votes)

- does anyone else speed the videos x2(24 votes)
- 1x1=1, ok now we will explain what that algebraically means in a 30 minute video(10 votes)

- I am confused when I use this in word problems.(20 votes)
- 5 Years Later...

I know how you feel. Believe me, when I just started, I thought I'd be old and gray before I finished! But when you get used to them, it actually isn't that hard.(14 votes)

- it sort of helps but im still confused(9 votes)
- Here's a tip: Start out by doing what's in the parenthesis. Then follow the rest of the rules in PEMDAS. If you still need help, just ask and I will help you with the problem.(26 votes)

- how do you do it if it is a negative minus aa number times a positive minus three

example -c-6(2c-6)

please answer(8 votes)- Hi again!

(answer to your question from Comments on my original reply)**Yes! we can**.*multiply*unlike terms

We can*multiply the coefficients*, then place the variables together, behind and against our new coefficient.

2x • 5y =**10xy**…

Why? The concepts in action

The**coefficient is the number**that is smooshed**in front of a variable**, the coefficient is**the count of that variable's repeats combined**, so it represents**the number**.)*of times*the variable occurs

(x + x) • (y + y + y + y + y)

=

2x • 5y

So…

★ That's one of the reasons we can multiply Unlike Terms is because**the relationship between the coefficient and the variable is**!*Multiplication*

Let's take a closer look…

2x • 5y

=

2 • x • 5 • y

★*Another reason*we can is the useful**Property of Multiplication**, (a truth about it), it's(interchangeable), which means we can**Commutative****rearrange****multiplication terms, and still get the**!*same answer*

4 • 3 • 2 = 24 = 2 • 4 • 3 = 24

Which is what we're doing (when we multiply unlike terms), just*rearranging a multiplication expression*, by**completing the calculation for the values we do know**(*the coefficients*),**and keeping the unknown values**(*variables*)**set up properly to multiply**!

2x • 5y

=

2 • x • 5 • y

=

2 • 5 • x • y

=

10 • x • y

=**10xy**←yay! 🥳**So that's why when**, we can*multiplying*unlike terms**just multiply the coefficients, and tack the variables on at the end**!

It works when there are exponents >1 as well, but take note of*addition or subtraction operations zones*, they act like barriers for the multiplication, only cross them after you complete the multiplication and*only if you can combine like terms*!

2ab • 4w^2 + 5x • 3y - z^2 • 5u

=

(2ab • 4w^2) + (5x • 3y) - (z^2 • 5u)

=

8abw^2 + 15xy - 5uz^2 ←yay! 🥳

We rarely need to*show*all the details, but knowing and understanding the 'how' and 'why' of 'what' is happening can help us take the next step up in comprehension!

I hope this helps someone! (≧▽≦)(23 votes)

- TO ANY ONE WHO IS CONFUSED ON THE DIFFERENCE BETWEEN TERMS AND EXPRESSIONS, I CAN HELP OUT:

Hey guys, HotCocoa here! I had trouble with this at first, too.

Expressions are like this simple one: 2+4, or -3 + (5 x 8v) x 29v. You see, it doesn't have an equal sign, it doesn't have an answer, it's not meant to be solved. now, an EQUATION is meant to be solved, like this: 99-37=?

Please like if this helped you, and comment if you have any more questions or want to add on or give me feedback.

Happy to help :)(12 votes) - i love kahn academy and i appreciate the help but sometimes... i think sal took a knife to my brain with this math stuff.(12 votes)
- Just so yall know I read all of yalls bios. Every one of them. On every video.(8 votes)
- read yours too, anna. diogenes was a bit of a madlad but respect.(4 votes)

- I'm confused in large equations(7 votes)
- does anyone else love it when he gives you the answer to a test question(6 votes)
- yeah but I dont really like watching the vids(4 votes)

## Video transcript

- Let's get some practice identifying equivalent expressions. So I have an expression written here in yellow and then I have two more written in this light green color. And I want you to pause this video and see if you can figure out which of these expressions, and it's possible that neither of them are, which of these are equivalent to the one in yellow. So I'm assuming you've had a go at it. So the way I like to tackle it is just to simplify all of them as much as possible. So this one up here is clearly not that simplified. So let's distribute this two. So if I distribute the two, what does it become? This is equal to two times negative six-C, is negative 12-C. Two times positive three is positive six. And then we have plus four-C, plus four-C. And then we can simplify it further cause I have both of these terms that involve C. I have negative 12-C plus four-C. So what's that going to be? Negative 12 of something plus four of something is going to be negative eight of that something. So this is going to be equal to negative eight-C. So these two blue terms when I add them, I'm going to get negative eight-C. And then finally plus six. Plus six. Now just doing that that's exactly what this first green expression is. So this one is definitely, this one is definitely going to be equivalent. Now what about this one down here? Well to figure that out let's simplify it. So let's distribute the three. Three times negative four-C is negative 12-C. Three times positive two is positive six. So plus six and then we have the plus four-C over there. It's lookin' good. And then we can add the terms that involve C. Negative 12-C and four-C, you add those together, you're gonna get negative eight-C. Negative eight-C plus six. Plus six, which is exactly what these other ones are. So all of these, all of these expressions, are actually equivalent. This one, that one, and that one. Let's do another example. And just like the last time pause the video and see which of these two expressions, it could be both of them or it could be none of them, or it could be one of them. Which of them, if any, are equivalent to this yellow expression? Alright let's do it together. And like before let's just simplify it. So the first thing my brain wants to do is let's take the terms involving N and add those together. So negative six of something, in this case N, plus four of that something, in that case N. So negative six-N plus four-N that's gonna leave you with negative two of that something. You add the coefficients. Negative six plus four is negative two Ns. So we have negative two-N, and then plus negative 12, that's the same thing as just minus 12. So minus, minus 12. So I simplified our original expression. Let's see these ones, these down here. So if I distribute the four, if I distribute the four I get four times N is four-N. And then four times negative three is minus 12. And then we are going to subtract six-N. So minus six-N. So what does this give us? We get, let me get another color here. So we have four-N, I'm adding all the terms with N, minus six-N, that's gonna give us negative two-N. And then we have the minus 12. And then we have the minus 12. So this expression when I simplify it got me the exact same place as the first expression. So these two, these two are equivalent. This is equivalent to that. Now let's check this one out. So two, let me just distribute, let me just distribute the two. Two times two-N is four-N. And then two times negative six is negative 12. So this simplified to four-N minus 12 which is clearly different than negative two-N minus 12. So this one, this one, is not the same as the other two.