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### Course: 7th grade (Illustrative Mathematics) > Unit 6

Lesson 13: Lesson 21: Combining like terms (part 2)# Combining like terms with negative coefficients & distribution

We've learned about order of operations and combining like terms. Let's layer the distributive property on top of this. Created by Sal Khan.

## Want to join the conversation?

- What is the key to getting the answer to problems like these? I get lost so fast!(356 votes)
- The key realization here is to realize that we can't combine things that are different from one another. And you must also learn or master how to distribute so you can solve these problems

I know this is probably not such a good example, I'll improve my example as soon as I can think of a better one.

For instance we have 3 oranges and 8 apples, in all we have 11 fruits. let o=oranges, a=apples, f=fruits

Mathematically we can say that 3o+8a=11f

But we know that oranges and apples are both fruits so

3f+8f=11f(340 votes)

- I'm slightly confused, in the second example it says 7(3y - 5) - 2 (10 + 4 y), but he simplifies the last parentheses as -20 -8y......What happened to the plus symbol during the simplification?(114 votes)
- emassingill1,

Sal distributed the -2 over the quantity in the second parenthesis. In other words he multiplied -2 * (10 +4y), which you do by multiplying the -2 times EVERYTHING in the parentheses. So -2*10=-20 and -2*4y=-8y. Adding those together yields:

-20+ -8y = -20-8y.(92 votes)

- What is it called when you replace a number for a letter(58 votes)
- When you replace a number with a letter the letter is called a variable. Variables can be used to figure out questions like " If sally used 4 stars for a painting how many would she need for 79 paintings? Write an expression.

Number of painings: x

stars:y

X4=y

Hope this helped!(14 votes)

- At3:20to3:36, how come he did not do the inverse of that operational sign when it comes to subtracting 21y - 35 - 20 - 8y if the operation of the 8y is negative?(33 votes)
- Leah,

If I understand your question, this answer might help.

21y-35-20-8y is the same thing as

21y + (-35) +(-20) +(-8y)

and addition is commutative so we can chage the order to

21y +(-8y) + (-35) +(-20) Now the 21y + -8y is 13y for the same reason that 21 apples miuns 8 apples is 13 apples, so

13y + (-35) +(-20) Now the -35 and -20 can be added to be -55 so

13y-55 is the answer.

I hope that helps make it click for you.(37 votes)

- I'm kind of confused, in the second example it says 7(3y - 5) - 2 (10 + 4 y), but Sal simplifies the last parentheses as -20 -8y so I am confused What happened to the plus symbol during the simplification?(20 votes)
- A negative multiplied by a positive results in a negative. When you multiply -2 by 4y, the answer will be -8y.(10 votes)

- I was practicing combining like terms with negative coefficients & distribution, and in the video they distributed the 7 to -5 when I thought the "-" was for subtraction. My question is, how can you tell the difference between when a "-" is a negative and when it's for subtraction when there are no parentheses?(10 votes)
- There's no way to tell, because they are essentially the same thing. Adding a negative number, is the same as subtracting a positive number.

5 - 2 = 3

5 + (-2) = 3

5 - 2 = 5 + (-2)

Hope that helps :)(24 votes)

- dont 2 negetives equal a positive though?(10 votes)
- It depends on what you are doing...

If you multiply/divide 2 negatives, you get a positive: -2 (-3) = +6

If you add 2 negatives, you will get a negative: -2 + (-3) = -5

Hope this helps.(18 votes)

- I am getting really confused if the answer should be negative or not? and should I subtract or add? The signs seem like they are way different in the answer!(6 votes)
- Way different signs in the answer is fine if we follow the rules of math with negatives…

In Distribution…

★**Multiplying**

negative × positive = negative

positive × negative = negative**Mismatch signs = Negative**

while all…**Matching signs = Positive**

negative × negative = positive

positive × positive = positive

★Combine Like Terms**Positives and Negatives**.*are addition and subtraction*.

Like Terms*match in both variable and exponent*

So…

to**Combine Like Terms**means…**merge matching terms**, keeping the dominant ± sign

★an**Absolute Value**

is a number's*distance from zero*

and.**tells us which sign to keep**

|-42| = 42

|2| = 2

Whichever number is the**furthest from zero dominates****with its sign**.

2 - 42 = -40

-42 + 2 = -40

♪**Same sign Add and Keep it**♪…

-3 - 7 = -10

-5 - 5 = -10

♪**Different sign Subtract, keep the sign of largest Absolute Value**, then you'll be exact. ♪

-43 + 1 = -42

1 - 43 = -42

★So…

•**Matching signs Add***same signs stay the same*

-33 - 300 = -333

while…

•**Mismatch signs Subtract***absolute value tells dominant sign***2 - 22***is like*…

-|22 - 2| = -20

or

- (22 - 2) = -20*because*…

|-22| > |+2|**Twenty-two spaces**from zero

>**is greater than**from zero

Two spaces*we keep the Negative sign*

3 - 36 = -33

-36 + 3 = -33

★Like competing in a…**Tug of War game over Origin**,*sign furthest from zero wins*,**by the answer being on its side of Origin**.

-10 + 1 = -9

=

1 - 10 = -9**yanked***TEN to the Negative left*

then**pulled***ONE to the Positive right*

=**Negative sign wins**with nine spaces still Left of Origin!

★Notice that because…*multiplication is repeat addition*

and**Negatives and Subtraction signs**, and Positives*are the same thing**are Addition signs*…

there's a**shared pattern**to which sign is correct…

In multiplication/division:**Matching signs = + Positive**

Mismatching signs = - Negative

In combining terms:**Matching signs + Add and Keep**

Mismatch signs - Subtract and Compete

(≧▽≦) Hope this helps!(23 votes)

- For anyone who doesn't understand why negative times negative equals to positive, I have an example to show you:

When I say "Eat!" I am encouraging you to eat (positive) But when I say "Do not eat!" I am saying the opposite (negative). Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive). So, two negatives make a positive.

Hope this helps:)(17 votes) - Is there any time where you don't turn the equation into its simplest form(14 votes)
- Yes and no, You can not have to simplify it, but it does not only make problems more easier and more easy, but also lets you solve the equation. Correct me if im wrong.(4 votes)

## Video transcript

I've gotten feedback
that all the Chuck Norris imagery in the last video might
have been a little bit too overwhelming. So for this video, I've included
something a little bit more soothing. Let's try to simplify
some more expressions. And we'll see we're
just applying ideas that we already knew about. Let's say I want to simplify the
expression 2 times 3x plus 5. Well, this literally
means two 3x plus 5's. So this is the exact same thing. This is one 3x plus
5, and then to that, I'm going to add
another 3x plus 5. This is literally what
2 times 3x plus 5 means. Well, this is the same thing
as, if we just look at it right over here, we have now two 3x's. So we could write
it as 2 times 3x. Plus, we have two 5's,
so plus 2 times 5. You might say, hey, Sal, isn't
this just the distributive property that I know
from arithmetic? I've essentially just
distributed the 2? 2 times 3x plus 2 times 5? And I would tell
you, yes, it is. And the whole reason why I'm
doing this is just to show you that it is exactly
what you already know. But with that out of the way,
let's continue to simplify it. When you multiply the 2
times the 3x, you get 6x. When you multiply the 2
times the 5, you get 10. So this simplified
to 6x plus 10. Now let's try something that's
a little bit more involved. Once again, really just
things that you already know. Let's say I had 7 times 3y minus
5 minus 2 times 10 plus 4y. Let's see if we
can simplify this. Well, let's work on
the left-hand side of the expression, the
7 times 3y minus 5. We just have to
distribute the 7. This is going to be 7 times 3y,
which is going to give us 21y. Or if I had 3 y's
7 times, that's going to be 21 y's, either way
you want to think about it. And then I have 7
times-- we've got to be careful with
the sign-- negative 5. 7 times negative
5 is negative 35. So we've simplified
this part of it. Let's simplify the
right-hand side. You might be tempted to say,
oh, 2 times 10 and 2 times 4y and then subtract them. And if you do that right and
you distribute the subtraction, it would work out. But I like think of
this as negative 2, and we're going to
distribute the negative 2 times 10 and the
negative 2 times 4y. So negative 2 times
10 is negative 20, so it's minus 20
right over here. And then negative 2 times
4, negative 2 times 4 is negative 8, so it's
going to be negative 8y. Let's write a minus
8y right over here. And are we done simplifying? Well, no, there's a little
bit more that we can do. We can't add the 21y to the
negative 35 or the negative 20 because these are
adding different things or subtracting different things. But we do have two things
that are multiplying y. Let me do all in
this green color. You have 21 y's right over here. And then we can view it as from
that we are subtracting 8 y's. So if I have 21 of something
and I take 8 of them away, I'm left with 13
of that something. So those are going to
simplify to 13 y's. I'll do this in a new color. And then I have
negative 35 minus 20. That's just going to
simplify to negative 55. So this whole thing
simplified, using a little bit of the distributive
property and combining similar or like terms,
we got to 13y minus 55.