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# 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! •   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
• 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? •   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.
• What is it called when you replace a number for a letter • At to , 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? •  Leah,
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

I hope that helps make it click for you.
• 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? • • 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? • Is there any time where you don't turn the equation into its simplest form • 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! • 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…
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
Two spaces
from zero
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…
and
Negatives and Subtraction signs are the same thing, and Positives 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!
• 