# Justify steps w/ property, definition, or operation: (3x + 10x) to x(3 + 10) to x(13)

#### mathdaughter

##### New member
In the following, justify each step with a property, definition or an operation.

1. Justify going from (3x + 10x) + 2y

to x(3 + 10) + 2y

We have two different answers: one says it is a result of a distributive operation, and another says it is a result of an associative operation.

which one is correct, distributive or associative?

2. Justify going from x(13) + 2y

to 13x + 2y

Which one is correct, distributive or commutative?

thanks!

Last edited by a moderator:

#### tkhunny

##### Moderator
Staff member
Associative involves only one function type.
Distributive actually has a much longer name: "The Distributive Property of Multiplication over Addition"

#### mathdaughter

##### New member
Associative involves only one function type.
Distributive actually has a much longer name: "The Distributive Property of Multiplication over Addition"
are you saying the answer for the first one is Associative and the answer for the second step is Distributive?

#### tkhunny

##### Moderator
Staff member
from (3x + 10x) + 2y
to x(3 + 10) + 2y
Are there TWO different operations in x (3 + 10) = 3x + 10x, or just one type of operation? Which of multiplication and addition do you see?

from x(13) + 2y
to 13x + 2y
Are there TWO different operations in x(13) = 13x, or just one type of operation? Which of multiplication and addition do you see?

#### mathdaughter

##### New member
Are there TWO different operations in x (3 + 10) = 3x + 10x, or just one type of operation? Which of multiplication and addition do you see?
There are two different operations: two multiplications and one addition. so it is not associative?

Are there TWO different operations in x(13) = 13x, or just one type of operation? Which of multiplication and addition do you see?
There is only one operation: multiplication.

#### mathdaughter

##### New member
Are there TWO different operations in x (3 + 10) = 3x + 10x, or just one type of operation? Which of multiplication and addition do you see?
we are talking about 3x + 10x = x (3 + 10) changes, not x (3 + 10) = 3x + 10x changes, right?

#### tkhunny

##### Moderator
Staff member
If you see multiplication and addition, it must be Distributive. The Distributive Property of Multiplication over addition.

3x + 10x = (3+10)x

If you see ONLY addition or ONLY multiplication, it CANNOT be Distributive.

10 + 3 = 3 + 10 -- Only addition
10*3 = 3*10 -- Only Multiplication

#### tkhunny

##### Moderator
Staff member
we are talking about 3x + 10x = x (3 + 10) changes, not x (3 + 10) = 3x + 10x changes, right?
Both of these are examples of the Distributive Property of Multiplication over Addition.

#### mathdaughter

##### New member
Both of these are examples of the Distributive Property of Multiplication over Addition.
I thought adding bracket is associative and removing bracket is distributive. thank you for your explanation.

#### JeffM

##### Elite Member
I thought adding bracket is associative and removing bracket is distributive. thank you for your explanation.
Mechanically, the associative properties of addition and multiplication are about switching brackets around, not changing the number of brackets.

$$\displaystyle (a + b) + c \equiv a + (b + c)$$ ASSOCIATIVE PROPERTY OF ADDITION.

Notice that no multiplication is involved and the number of brackets does not change.

$$\displaystyle (a * b) * c \equiv a * (b * c)$$ ASSOCIATIVE PROPERTY OF MULTIPLICATION.

Notice that no addition is involved and the number of brackets does not change.

Essentially the associative property of addition says that if you need to add up
3, 7, and 11, it makes no difference if you add 3 and 7 to get 10 and then add 10 and 11 to get 21 or if instead you add 7 and 11 to get 18 and then add 18 and 3 to get 21 because the answer is the same either way.

Similarly, if you need to multiply 3, 7, and 11, you get the same answer of 231 whether you first multiply 3 and 7 to get 21 and then multiply 21 by 11 to get 231 or whether you first multiply 7 and 11 to get 77 and then multiply 77 by 3 to get 231.

As tkhunny has explained, the associative and commutative properties of addition do not have anything to do with multiplication, and the associative and commutative properties of multiplication do not have anything to do with addition. The distributive property is the only one that deals with both multiplication and addition at the same time.

$$\displaystyle a * (b + c) \equiv (a * b) + (b * c)$$

DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION.

What this one means is that the product of 3 and the sum of 7 and 11 (meaning 3 times 18 or 54 in total) is the same as the sum of the products of 3 and 7 and of 3 and 11 (meaning the sum of 21 and 33 or 54 in total).

The arithmetic facts described by these properties are so obvious that students may think that they are missing some subtlety and get confused about merely naming things that they have known since second grade.

Last edited:

#### mathdaughter

##### New member
All of you are really helpful.

#### mmm4444bot

##### Super Moderator
Staff member
I thought adding bracket is associative …
I have seen the Associative Property used to add brackets to an expression that didn't have any before. How this property is described or used sometimes depends on the instructor. :cool: