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### Course: Class 7 (Old)>Unit 10

Lesson 2: Combining like terms

# Combining like terms with negative coefficients

This example of combining like terms in an expression gets a little hairy. Pay attention. Created by Sal Khan.

## Want to join the conversation?

• So x's cannot be combined with xy's or with x^2's, and y's cannot be combined with xy's or with y^2's, right? Doesn't this leave your equation containing a lot of terms still? Wouldn't it be a lot simpler if we could reduce the number of terms more?
• The way I think about it, if x = Turkey and y = farm, then 2 x (Turkeys) are not 3y (Farms) and close but not quite 4xy (Turkey Farms) xy. I just went back to the earlier videos and substituted variables for objects to make myself understand it.
• How do you recognize the difference between the 'minus' sign and the 'negative' sign.
I'm so confused
• Think of them as being the same thing, it is one or the other based on where it is in the equation. The negative sign is used at the beginning of something new, so it could be primarily in three places -3(-4 + x)/-6. A negative is beginning of an expression, the beginning of a parentheses, and the beginning of the denominator. A minus is between things such as 3(2x - 5) - 5(4x - 2) . In this case, it is between an x and a constant and between two expressions. In this case, if you try to distribute -5(4x-2) independently, the minus now is at the beginning and so it is considered a negative sign rather than the minus it was in the full expression. Does this help?
• I have been having trouble understand the -5+6=? I don't understand last video he had two numbers with - signs and he added them and kept the sign why? What am I missing maybe video that could help it seems so simple but I have no clue! Any feed back will be welcomed.
• I think the easiest way to understand this is via the number line.
When we add we move to the right along the number line, and when we subtract we move to the left.
Starting at 0 (zero) we move 5 steps to the left and from there we move 6 steps to the right.

` −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 `
`−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−+−−`
` <−−−−( −5 )−−−−− `
` −−−−−−( +6 )−−−−−−> `

As you can see, this lands us at 1.
So, −5 + 6 = 1

– – –

Adding two negative numbers, for example (−3) + (−4), is the same as (from 0) moving 3 steps to the left and then another 4 steps to the left, in total 7 steps to the left, which lands us on (−7),
and we write −3 − 4 = −(3 + 4) = −7

– – –

Sal explains it more thoroughly here:
• at why did he leave out the y2 shouldn't that count as a y to?
• if y=3, then y2 would be 9, so he can't add them together
• At 0.21, what is the difference between x^2 and 2x?
• Exponents are shorthand for repetitive multiplication. So `x^2` = `x * x`
Multiplication is shorthand for repetitive addition: `2x` = `x + x`
Hope this helps.
• Im confused, please whoever explains how to do it please do so thoroughly, that i may understand.

Please explain: Combining like terms with negative coefficient
• I'm unsure what exactly you are confused about, so I will try to explain the entire concept as thoroughly as possible. If I don't cover what you are confused on, though, just tell me and I can explain that.

First, let's set up an example problem:

2y + 3xy + 4x - 5x^2 -7y + xy +x^2

Remember that a term is any coefficient and any variable. So, as an example, 3x, 7y, 9z^2, and 4f are all terms. Before we do anything to combine these terms, we need to remember the rules for what terms to combine. We can only combine terms that have the same variable. That means we could combine an x with a 3x, because x is the variable in both of those terms, but we cannot combine a 4x with a 2xy. Even though both terms share an x, an xy is not the same as an x. In addition to multiplying by x, you are also multiplying by y. In short, the variables following the coefficients need to be identical in order to combine them.

Let's recap our example

2y + 3xy + 4x - 5x^2 -7y + xy +x^2

Knowing what was just covered, let's rearrange the terms so that all terms with identical variables are next to each other. When doing this, remember the sign that the term carries. For example, if we wanted to rearrange x - 3xy, we would write it as -3xy + x because subtracting a term is the same as that term being negative. It helps to think of the + or - sign (and only + or - signs) in front of a term as being a part of the term itself. With that out of the way, our example becomes

2y - 7y + 3xy +xy + 4x -5x^2 + x^2

Now, we combine our terms. Remember that when we combine terms to think of it like combining the coefficients. No matter what numbers we combine, our variable will always stay the same. For example, the z in 3z + 9z will stay z, even after we combine the coefficients to get 12. Starting with the y variables, we have 2y - 7y. Here, we simply subtract. If you are unsure how to do this, that's okay! Don't worry. I would just go back and review negative numbers before coming back to this topic. It is important to have a good foundation. Continuing on, combining those terms gets us

-5y

Which added to the rest of the expression would be

-5y + 3xy +xy + 4x -5x^2 + x^2

Now we can combine the xy terms. 3xy and xy. Remember that there is an imaginary 1 in front of the xy. So, we can visualize the combination of these terms like 3+1. Now we have

-5y + 4xy + 4x -5x^2 + x^2

Our 4x is the only term with an x variable, so we leave that alone. Finally, we can combine our x^2 terms.

-5x^2 and x^2

This will give us -4x^2 (remember that the ^2 is a part of our variable, and because of that it will not change after we combine our coefficients), leaving us with

-5y + 4xy + 4x -4x^2

I have had teachers that were picky about having the terms being ordered in descending order, from largest exponent to smallest, so I would rearrange it like

-4x^2 +4x -5y +4xy

There we have it! We successfully combined like terms with negative coefficients. Once again, I hope my explanation was thorough enough, but feel free to reach out if it wasn't. Best of luck on your learning adventures, and remember that unlike y= mx +b, progress isn't linear. Don't let setbacks like these decrease your confidence and motivation. :)
• HELP?
On some examples it shows that we move the numbers on the other side of the equal = sign... I get that once thats done the numbers either become a negative or positive but... how do we know when to use it. There's some questions I can solve using the same method in this video but then there's this eg:
-3n - 7 + (-6n) + 1
If I were to use the method above would it be written as so?
-3n - (-6n) -7 + 1 ?
In this equation they used
-3n - 7 + (-6n) + 1
= (3+6)n -7 +1
I don't get it? which method do we use? if both... how are we supposed to know which one is right? or when to use either one? Please.
• First, you need to recognize an equation vs an expression.
An expression contains no equals sign to start with. All the examples in this video (and your example in your question) are expressions. An equation is made up of 2 expressions separated by an equals sign.

We only change the signs to move terms across an equals sign. This requires that you have an equation. When you changed: -3n - 7 + (-6n) + 1 into -3n - (-6n) -7 + 1, you moved the "-(-6n)" to the other side of the -7. You did not more it across and equals sign. So you do not change the sign.

Hope this helps.
• How do I type exponents? like I want to type the numbers "three times x squared", how might that be done? I was struggling doing it with Khanmigo.
• Type: 3x^2
The "^" symbol denotes that the number that follows is an exponent.
• how do you recognize the difference between a negative sign and a minus? it is all very confusing.
• When combining like terms with negative coefficients, negative signs and subtraction signs mean the same thing, and positive signs and addition signs are the same as well.

★Actually, they always have, through out all of addition and subtraction plus and minus signs, + and - have always meant positive and negative number movement.

3 - 2 = 1
means positive 3 combined with negative 2 equals positive 1.

+3 combined with -2 = +1

So we've been doing this same math all along
Just now, at this stage, we're using a broader spread of the existing number values placed in combinations that weren't given to us before.

-5 - 7 = -12
Means to combine a negative 5 and a negative 7, the subtraction sign belongs to the seven, creating a negative 7.

So…
-5 - 7 = -12
means…
-5 + -7 = -12

This is also written as:
-5 + (-7) = -12

• If we intend to subtract a negative value, we must show it by using parentheses.

-5 - (-7) = 2

The notation of 'negative parentheses negative', separates the -7 in a subtraction problem, because it's considered bad form to write the subtraction of a negative with consecutive minus signs.

-5 - -7 = 2 ←nope, bad form

In this -(- situation)…
The negative signs on each side of the parentheses cancel out.
-(-7) is just like an invisible negative one in front of parentheses that needs to be distributed, all signs within the parentheses get flipped to their opposite.

-5 - (-7) = 2
=
-5 + (+7) = 2
=
-5 + 7 = 2

★Combining Like Terms means to merge the known values, (coefficients), that have the same variable and exponent.

To merge (both positive and negative) matching terms to combine them into one term.

★ So for this level, we're just learning addition and subtraction of any number value combination!

(≧▽≦) I hope this helps!