Arithmetic properties

# The distributive property

## Distributive property algebraic expressions

Here we have some algebraic expressions to which we need to apply the distributive property. Now we're beginning to see how useful this property can be!
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## Distributive property algebraic expressions

Discussion and questions for this video
When you did problem c. 6+(x-5)+7. Why didn't you perform distributive property?
There is no distribution taking place in this problem because everything is addition; no multiplication at all. Or if you'd like to think about it this way, there is a 1 hiding in front of the left parenthese, so distribute the 1 and you get 6 + x - 5 + 7.
whats with the m c x y stuff i dont get it
m c y x are all variables used in math, but they are most commonly used in algebra.
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in question c) .. 11/2 we don't divide them ?
You could, and get 5 1/2 (5.5), but sometimes its better to just have an improper fraction.
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How do you remember the differences between commutative, associative and distributive law of addition ? That is, without mixing them all up!!
I just read it over and over again until I memorised it. Effort!
I am 'struggling' on the Rational Number Word Problems test and this question has me completely confused. I don't understand the part where it says we simplify d - 0.035 * d to d(1-1.035) using the distributive property. How do we just drop a 'd' and earn a '1'? I pasted the whole word problem with the first few hints below.

The earth's orbit around the sun is not a perfect circle. Let's call the smallest distance from the earth to the sun d. Earth's farthest point from the sun is equal to 1.035⋅d. If the distance from the earth to the moon is currently 0.0016⋅d, how many times larger is the distance between earth's farthest and closest point than earth's distance to the moon? (Express your answer as a simple fraction in lowest terms)

We first need to find the distance between earth's closest point to the sun, d, and earth's farthest point from the sun, 1.035⋅d. How do we find this?
To find the distance between d and 1.035⋅d, we need to compute the absolute value of their difference. What is |d−1.035⋅d| equal to?
Using the distributive property, we see that the difference d−1.035⋅d is equal to d⋅(1−1.035). Since 1−1.035=−0.035, this is simply equal to −0.035⋅d. What is the absolute value of −0.035⋅d, and how do we use it to get the answer?
I'm going to assume that you do know the distributive property. So when we have d(1-1.035) and we distribute to both of the terms, we end up with d - 1.035d.
What is unique about the distributive property though, is that it works in both directions. Essentially since we know that d(1-1.035) = d - 1.035d, we also know that it works in the other direction, so d - 1.035d = d(1-1.035). What really happened is, we divided each term by d so that the d was on the outside of the parentheses. d/d = 1. and -1.035d/d = -1.035. Putting these all together gets d(1-1.035). I hope I answered your question. If not, I apologize.
i am a 5th grader going into 6th next year, and i was wondering if someone can answer this question:
do you think i am old enough to be practicing distributive property, because i have to do this unit on mult. and div. and i am wondering if this is expected of me. i do not know how to work with negative numbers, which makes this all even more confusin than it already is.
I think you're at a good age to learn the distributive property. But, you will have to know how to work with negative numbers, so you should learn that before you go on to the distributive property. Hope I helped :)
i didnt understand his example: E. 2:31
hmmmm....I don't know if I can explain it better, but I'll try...ok the problem is 4(m+7)-6(4-m). Now to distribute you start by simplifying...4(m+7) is the same as 4 x m +7...but the distributive property lets you multiply both of them, like so...4m + 28. Now your problem looks like this...4m + 28 - 6(4-m)... Now you do the same thing here as you did with the first. So now your problem looks like this... 4m + 28 - 24 + 6m...Next you combine like terms...6m + 4m = 10m...28-24 = 4...So your answer is 10m + 4

Hope that helps!
1 Comment
when your doing e, why do you put parenthesis around 24-6m but you don't put it for 4m+28 ?
because he didn't distribute the -1 into the quantity, Just the +6.
Could have just said -24+6m without parenthesis.
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At 1:34, example c, why is it possible to just remove the parentheses and go on like there haven't been ones? Aren't they here for a reason? To calculate x-5 first, meaning before 5-7? I don't understand!
1 Comment
There's not necessarily a reason; sometimes they just put parentheses to trip you up in exams, to see if you know your operation rules. One of those rules is: addition is commutative. It doesn't matter if you do 3 + 5 + 6 or 5 + 3 + 6, you'll get the same result.

Subtraction is the same as adding a negative number, so it's commutative too. Hence you don't need the parentheses in 6 + (x - 5) + 7, and can do the operation in the order you want, as long as you make sure the negative sign stays with the 5. What I mean is, you can't go "OK, let's do 6 + 5 + x - 7". But other than that, you can do what you want!

Lastly, as I said, subtraction is commutative, which means you can change the order of the terms and it won't change the result. But you might say "that's not true, 6 - 8 isn't the same as 8 - 6". To which I'd reply: no, they aren't the same BECAUSE you put the negative sign on another number. As I said above, subtraction is like adding a negative number; that means that 6 - 8 is really 6 + -8. You can change the order, but no matter what you do, 8 will have to stay negative. You can't just make 6 negative instead, that's not part of the commutative law! So the only way to rewrite 6 - 8 is -8 + 6.

Hope it helps!
What does all that a b and c mean? I know that they are variables, but does he have to use them? Please help!
He doesn't have to. But it's a good idea to get comfortable to seeing different letters other than "X" and "Y" used as variables. In science classes, you'll need to use different letters to keep track of everything.

Here are some common letters that are used as variables:
t, for time
d, for distance
v, for velocity (velocity is like speed, but you have to pay attention to where you're going)
a, for acceleration (how fast you're speeding up)
where did you get the -1? at exercise e).
Because when you have (Example) -(4x-5) is the same as (-1)(4x-5), so in the question that Sal does its: -(24-6m) is the same as -1(24-6m) or (-1)(24-6m). This causes you to change all the signs within the brackets.
How would you be able to solve problems like 4(8n+2) ? Or at least point me to the section where I can find videos to solve distributive property equations like this one. Please >_<
To solve this just think of what it is telling you to do by how it is written. In words, it would be: multiply everything inside the parentheses by 4.
So, 4 x 8n = 32n and 4 x 2 = 8. Now that you have the distribution done you have the answer:
32n + 8
Hope that helps.
I don't understand the importance of the distributive property. Could someone explain it to me? Also, what kind of career uses the distributive property?
The distributive property, when manipulated to its fullest, can be extremely useful. Once you get into Algebra, you will learn FOIL, which is an extension of the distributive property that lets you multiply (a+b) times (c+d) to a(c+d)+b(c+d) to ac+ad+bc+bd. Once you do that, you can do the opposite and bring ac+ad+bc+bd back to (a+b)(c+d). This will help you solve equations with x^2 in them [you will factor from x^2+(a+b)x+ab into (x+a)(x+b) instead of (a+b)(c+d)], which can help you find where thrown objects will land on the ground by approximating their path using functions with x^2 in them.

However, that's much for a much later time. The uses of this simple distributive property are more daily-life than the above example given. For example, if you know that the number of bottles you must distribute to 3 children and the number of bottles you must distribute to 7 children is 40 and the number of bottles per student is the same, you can set up the equation 3a+7a=40. With the distributive property, you know 3a+7a=a(3+7)=10a, so 10a=40, or a=4. Adding like terms (adding terms with the same variables) is used many times in math and you'll find it helpful in careers.

I hope this helps!
is ) I* IO \$ @ ()*&(87978(*7(78(*79*&9 can they all be variables?
All those can be variables.
A variable can be any number or symbol.
could someone tell me why, when I do -5 x -11 it comes out -16?
-5x - 11 is as far as it goes. -5x - 11 is NOT equal to -5 -11 (that would be equal to -16 btw).

5x is a term and - 11 is an integer. Apples and oranges. There is nothing further to simplify.
Whats the difference between associative, communitive and distributive.
Commutative-The order of the numbers in multiplication or addition doesn't matter.
Examples:
1+2=2+1
27*8=8*27

Associative-The placement of parentheses doesn't matter if there is a close parentheses for every open parentheses and the operations are either all addition or all multiplication.
Examples:
(27+38)+9=27+(38+9)
(6*5)**4=6**(5*4)

Distributive-A number multiplied by two or more addends together is that number multiplied by each addend separately.
Examples:
3*(47+37+3)=3*47+3*37+3*3
4*(2-1+3)=4*(2+(-1)+3)=4*2+4*-1+4*3=4*2-4*1+4*3
6*(56-34)=6*56-6*34

The commutative property has to do with moving numbers around in addition and multiplication, the associative property has to do with moving parentheses around in all addition and all multiplication, and the distributive property has to do with distributing a multiplied number to many addends.

I hope this helps!
On problem E, you will be using the distributive property twice. After you do 4(m+7) and distribute that, go on to the next part of this problem. Since a subtraction sign is in front of the 6, problem would now be -6(4-m). Once you finish distributing in that part of the problem, you would put all of your answers into a problem. That would become 4m+28-24+6m. We would use a subtraction sign in between 28 and 24 because we distributed with a -6 and the answer to -6*4 would equal -24. From there, you would combine the like terms. 4m and 6m are like terms. Since there is an addition sign before the 6m, you would do 4m+6m which equals 10m. 28 and 24 are also like terms. Since there is a subtraction sign before the 24, you would do 28-24. That would become 10m+4. You combine like terms because it makes the problem simpler. Combining like terms will be a way to get the simplest form of an answer. For more on combining terms, look up Combine Like Terms on Khan Academy. I hope this helps.
Two negatives become a positive abd you keep the variable.
At 1:50 it looks like the problem is 6(x (MINUS) 5)+7, but Sal turns it into 6(x (NEGATIVE) 5)+7. Can someone please explain why this happens/happened?
but if it is a minus, won't it become
6 + x - 12 ?
if it is a negative it becomes
6 + x. 2
I am confuse on this part too
1:14 what does sampling mean?
Sampling means a small quantity to show the entire basis of the thing you are trying to find what it is like
How do you solve this?
-5(y+5)=15y
This is what I did
-5 times y equals -5y+ -25=15y
but then do i add or subtract the 25 to both sides?
1) First distribute: -5 (y+5) = 15y -> -5y - 25 = 15y.
2) Then get y by itself by subtracting -5y (or adding 5y) from both sides:
-25 = 20y.
3) Then divide both sides by 20. -25/20 = 20y/20 -> y = -5/4.
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I always struggle with distributive property. It seems like I always get different results when I attempt new problems I have never seen. Is there an 'order' to distributing - like order of operations?
No there isn't a order on doing it but it is still complicating.So I agree with you.
wht is six -x
6x simply means 6 times a number that we don't know. We represent that number we don't know with a _variable_. A variable is a letter or symbol to represent a *number*.
Give a thumbs up SALS AWESOME. He helps with everything. Anyway, isn't a variable only X and not Y or M.

Thanks
anything could be a variable. a variable represents and unknown number
Did anybody see the opening ceremony for the london 2012 olympics.
Can Sal please make a video on reverse distributive property?
Just look for the _*greatest common factor*_, that's all you need to do, I promise! ;-)

It's sometimes called _*factoring*_ ( or "_*taking out*_" ) , that's why you initially might not have found it.
There's actually a couple of these videos on Khan Academy!
is sunlight matter
Yes, sunlight is matter because anything and everything that takes up space is matter. Good job on figuring that out!

Except some people say that sunlight doesn't take up space and some people say it does, so it may or may not be matter...
I don't under stand what the fancy x means. Can anyone help me?
I think he is just using it as a variable, to express an unknown, or part of a mathematical property.
why @ 4:37 is 11/2x and not 5.5x he divides the 12. why not the 11?
You could write 5.5x + 6 and it would be equivalent to what he does. However, in algebra the convention is to simplify a fraction and leave it at that instead of using decimals. 12/2 simplified becomes 6/1 which is written as 6. Also a convention.
Some teachers will mark you down when you turn a fraction into a decimal when the problem states to simplify the fraction.
What if there are no numbers
so like X(Y+Z)
When you did problem c. 6+(x-5)+7. Why didn't you perform distributive property?in question c) .. 11/2 we don't divide them ?
Why do you sometimes change the addition or subtraction to its opposite?
ian zhong is my bro. He is asian and very smart
Is there a way to flag a video for future viewing? I am thinking that this will become more relevant as I progress further. I was lost with fractions and other operations.
Sharon, you are correct. Distribution is something you will use constantly as you progress into more complex math problems. Actually, it's a very easy concept and you'll have a lot of fun with it! I would suggest that if you have had problems with other things like fractions, go to your exercise dashboard and do drills over and over and over. Just like anything else, practice makes perfect. Or close to it. I've found if I run up against something that stumps me, it means I haven't perfected my understanding of something that came before. Keep me posted!
for the problem 11x+12/2 would the answer still be right if I put 5 1/2x+6?
Given:
11x + 12/2
Division property of equality:
11x + 6

The answer to your problem is 11x + 6 not 5 1/2x + 6
Wat is up with e?
At 4:40, why did he skip over b?
He just wanted to show us the other ones thats why he skiped letter d
why do we get a positive number when we multiply tow negative numbers?
Think of the negative on the first factor as meaning "opposite". Then keep in mind that multiplication means repeated addition. So you are finding the opposite of a negative number when you multiply two negatives. Here is an example to illustrate what I mean:
-3 x -4 = the opposite of 3 x -4 or the opposite of -4+ -4+ -4 which means the opposite of -12. The opposite of -12 is 12. That is why -3 x -4 = 12
Sorry for the amount of questions. But I need help. if 16 (5/8 - 1/4) is 6 for a final answer, how does one get there? Like how does one work through the problem in order to get this answer. Thanks y'all!
In order to fully grasp the distributive property first,
1) do it by doing the subtraction in parenthesis first then multiplying.
2) Do it by doing 16 * 5/8 - 16 * 1/4
so you put sand b together (ab) so thats multiplication right?Couldn't you just put a multiplication sign or a dot?
Because using variables, x being one of the most popular, will make people confused if you have something multiplied with x, for example, ax instead of axx. The dot is up to preference.
A negative multiplied by a negative is always a positive.
Yes, you are correct. :)
I still don't get the difference between x and y
1 Comment
x = Independent Variable
y = Dependent Variable
The value of y depends on the value of x.
Is this correct?
In the first series of questions:
(b) 0.6(0.2x+0.7)
=0.6*0.2x+0.7
=0.12x+0.7

(d) 6-(x-5)+7
=6+7-(x-5)
=13-x-5
=8+x

(f) -5(y-11)+2y
=-5y-(-55)+2y
=-5y+55+2y
=-5y+2y+55
=-3y+55

Sorry I'm not good at math...
Thank you.
For the first question you must times your second term by 0.6 too. Also take care when multiplying a decimal - you need the same amount of numbers after the decimal point as in the two being multiplied - if both are one decimal point then need two decimal points in your answer.
Eg. =0.6*0.2x+0.6*0.7
=0.12x+0.42
The second question you must multiple the second term by -1 as well as the first -(x-7)= - x - -7= -x+7
What do the letters mean? I don't understand that.
they are variables .In this they are the unknown numbers
At 4:41 , Sal said that you divide 11x/3. What do you do if the division isn't even?
You have a decimal on 4:41
Okay so what if the problem is something like: -6x - 7(-5x + 8) = 118

Would you use the same process if there is two "x"'s? Do you combine them or multiply them?

Or am I in the completely wrong place to be asking these questions because to be honest I have no idea what my math homework is about.
The distributive property still applies because multiplication comes before addition/subtraction. You would get:
-6x+35x-56=118
29x=174
x=6
Why do they have to use letters instead of numbers? its very confusing
the letters represent numbers that we don't know yet.
At 0:49, why do we have to multiply first then ad? Wouldn't it be much easier to just ad then multiply?
Because of Order of Operations! (:
What is the answer for D?
6 - (x-5) + 7
Distribute the -1 with x and -5:
6 -x +5 +7
18 - x
do I use the distributive property for the problem (7)(m+3) to get 7m + 21 or because there's parenthesis around the 7 is it 7 x m + 3
You had it right the first time. Since the parentheses go around the 3 as well, it also gets multiplied by the 7.

(7)(m+3) = 7m+21
When will koalas finally conquer the world
Does this have anything to do with math??/
At the beginning of the video on his first example, did Sal define the variables so he could get an answer? You can't do that in algebraic expressions, though, can you?
Actually you can, you just can't arbitrarily decide to give a value to unknown variables. It might be easier for example rather than dragging an inexact number around for say sin 45 * cos 45 * tan 45 to say that some variable t = sin 45 * cos 45 * tan 45. With the understanding that I'm using t as a placeholder for that value, so I don't have to carry long decimal digits around, through each operation until the very end.
i'm not getting the m and the x thing
In mathematics letters [like m and x] are often used to represent numbers we either to not know what they are [yet] or do not care what they are [because it does not matter]
I don't understand where the negative 1 (-1) came from in the "e" question. Can somebody explain me?

I'm talking about the expression 4(m+7)-6(4-m)
With the expression - (24 - 6m), you have to distribute the minus sign to all parts in the parentheses. A good way to think of it is that there is really a 1 after the minus sign (which is true). So it can be rewritten as -1(24 - 6m). You don't need to put the 1 there, but if it helps you, then put it.
what does x,m,z,and y mean?
They are variables. They represent numbers of unknown value.
Simplify the following expression:

−2(9m+1)+7(−9−5m)

Distribute the −2 into the first set of parentheses:

−2(9m+1)+7(−9−5m)

−18m−2+7(−9−5m)

Distribute the 7 into the parentheses:

−18m−2+7(−9−5m)

−18m−2−63−35m

Rewrite the expression to group the m terms and numeric terms:

−18m−35m−2−63

Combine the m terms:

−53m−2−63

Combine the numeric terms:

−53m−65

The simplified expression is −53m−65.
This problem was in the practice section of the website.
I am confused about how the negative(subtraction), addition signs work. The answer I came up with was -53m+-65 and it told me I was wrong. Is that correct? The only problem I am having is with the addition subtraction(negative) signs. Please give an detailed answer.
what does x and y and m and all that mean?
When we have numbers that can be very different depending on the situation, (for example: Let's say that for every kid in class you need 3 pencils and 2 pens and 1 eraser, and you have to write a 'rule' into the computer program- you know, how many different stuff it should order for every class,)- we just call the numbers names, usually letters like x y or z, and the computer can add them up for every class separately!
If you get this video, it's just a matter of time and practice before you are acing calculus!!!
Why did they create this rule!
:-(
To make many problems that this rule can apply to easier to solve
At 5:16 how does it become 1/3?
At 5:04, Sal rewrites the fraction with a -1/3 in front. (He factors the -1/3 out) In effect, he was just rewriting the expression as multiplication of -1/3 and the numerator.