Subtractivity is not commutative. For example, 4 − 7 does not have actually the exact same distinction as 7 − 4. The − sign here means subtractivity.

You are watching: Name the property of real numbers illustrated by the equation. -4(x + 3) = -4x - 12

However before, respeak to that 4 − 7 can be recreated as 4 + (−7), considering that subtracting a number is the same as including its opposite. Applying the commutative residential property for addition right here, you can say that 4 + (−7) is the very same as (−7) + 4. Notice just how this expression is exceptionally different than 7 – 4.

Now look at some multiplication examples.


Original Equation

Recreated Equation

4.5 · 2 = 9

2 · 4.5 = 9

(−5) · 3 = -15

3 · (−5) = -15

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Commutative Property of Multiplication

For any type of real numbers a and b, a · b = b · a.

Order does not matter as lengthy as the two quantities are being multiplied together. This building works for actual numbers and also for variables that represent genuine numbers.

Just as subtraction is not commutative, neither is division commutative. 4 ÷ 2 does not have the exact same quotient as 2 ÷ 4.


Example

Problem

Write the expression (15.5) + 35.5 in a various means, using the commutative residential property of enhancement, and show that both expressions lead to the very same answer.

(−15.5) + 35.5 = 20

35.5 + (−15.5)

35.5 + (−15.5)

35.5 – 15.5 = 20

Adding.

Using the commutative building, you can switch the −15.5 and also the 35.5 so that they are in a different order.

Adding 35.5 and −15.5 is the same as subtracting 15.5 from 35.5. The sum is 20.

Answer (−15.5) + 35.5 = 20 and 35.5 + (−15.5) = 20


Rewrite 52 • y in a different means, making use of the commutative property of multiplication. Keep in mind that y represents a genuine number.

A) 5y • 2

B) 52y

C) 26 • 2 • y

 

D) y • 52


Show/Hide Answer

A) 5y • 2

Incorrect. You cannot switch one digit from 52 and also connect it to the variable y. The correct answer is y • 52.

B) 52y

Incorrect. This is one more way to rewrite 52 • y, yet the commutative residential or commercial property has actually not been supplied. The correct answer is y • 52.

C) 26 • 2 • y

Incorrect. You do not must aspect 52 into 26 · 2. The correct answer is y • 52.

D) y • 52

Correct. The order of components is reversed.

The Associative Properties of Addition and Multiplication


The associative home of addition states that numbers in an addition expression can be grouped in various methods without transforming the amount. You have the right to remember the meaning of the associative home by remembering that when you associate via family members, friends, and co-workers, you finish up developing groups with them.

Below, are two methods of simplifying the same addition trouble. In the initially example, 4 is grouped through 5, and 4 + 5 = 9.

4 + 5 + 6 = 9 + 6 = 15

Here, the exact same problem is operated by grouping 5 and also 6 initially, 5 + 6 = 11.

4 + 5 + 6 = 4 + 11 = 15

In both cases, the sum is the exact same. This illustrates that transforming the grouping of numbers once including yields the very same sum.

Mathematicians regularly use parentheses to suggest which operation must be done first in an algebraic equation. The enhancement troubles from over are recomposed right here, this time utilizing parentheses to suggest the associative grouping.

(4 + 5) + 6 = 9 + 6 = 15

4 + (5 + 6) = 4 + 11 = 15

It is clear that the parentheses carry out not affect the sum; the amount is the exact same regardmuch less of wbelow the parentheses are inserted.

Associative Property of Addition

For any type of real numbers a, b, and c, (a + b) + c = a + (b + c).

The example listed below shows just how the associative residential property have the right to be provided to simplify expressions via genuine numbers.


Example

Problem

Rewrite 7 + 2 + 8.5 – 3.5 in 2 different means using the associative property of addition. Sjust how that the expressions yield the same answer.

7 + 2 + 8.5 – 3.5

7 + 2 + 8.5 + (−3.5)

The associative residential property does not use to expressions entailing subtraction. So, re-compose the expression as enhancement of a negative number.

(7 + 2) + 8.5 + (−3.5)

9 + 8.5 + (−3.5)

17.5 + (−3.5)

17.5 – 3.5 = 14

Group 7 and also 2, and add them together. Then, add 8.5 to that sum. Finally, include −3.5, which is the exact same as subtracting 3.5.

Subtract 3.5. The amount is 14.

7 + 2 + (8.5 + (−3.5))

7 + 2 + 5

9 + 5

14

Group 8.5 and also –3.5, and also include them together to get 5. Then add 7 and 2, and include that sum to the 5.

The sum is 14.

Answer (7 + 2) + 8.5 – 3.5 = 14 and 7 + 2 + (8.5 + (−3.5)) = 14


Multiplication has actually an associative home that functions exactly the very same as the one for addition. The For 3 or even more genuine numbers, the product is the very same regardmuch less of just how you team the numbers. For instance, (3 • 5) • 7 = 3 • (5 • 7).


")">associative residential or commercial property of multiplication
claims that numbers in a multiplication expression deserve to be regrouped using parentheses. For example, the expression listed below have the right to be rewritten in 2 various means using the associative property.

Initial expression:

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Expression 1:

*

Expression 2:

*

 

The parentheses carry out not impact the product, the product is the very same regardless of wright here the parentheses are.

Associative Property of Multiplication

For any type of genuine numbers a, b, and c, (a • b) • c = a • (b • c).

Recompose

*
 utilizing only the associative residential or commercial property.

A)

B)

C)

D)


Show/Hide Answer

A)

Correct. Here, the numbers are regrouped. Now

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 and  are grouped in parentheses rather of  and also 6.

B)

Incorrect. The order of numbers is not readjusted as soon as you are rewriting the expression making use of the associative residential or commercial property of multiplication. How they are grouped need to change. The correct answer is .

C)  

Incorrect. The order of numbers is not adjusted as soon as you are rewriting the expression using the associative building of multiplication. Only exactly how they are grouped have to change. The correct answer is .

D)  

Incorrect. Multiplying within the parentheses is not an application of the property. The correct answer is .

Using the Associative and also Commutative Properties


You will uncover that the associative and commutative properties are useful devices in algebra, particularly as soon as you evaluate expressions. Using the commutative and associative properties, you can reorder terms in an expression so that compatible numbers are next to each various other and also grouped together. Compatible numbers are numbers that are simple for you to compute, such as 5 + 5, or 3 · 10, or 12 – 2, or 100 ÷ 20. (The main criteria for compatible numbers is that they “occupational well” together.) The two examples below show how this is done.


Example

Problem

Evaluate the expression 4 · (x · 27) once

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.

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Original expression.

Substitute  for x.

Use the associative home of multiplication to regroup the components so that 4 and also  are alongside each various other. Multiplying 4 by  initially makes the expression a little bit simpler to evaluate than multiplying  by 27.

Multiply. 4 times  = −3, and also −3 times 27 is −81.

Answer

*
 as soon as
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.


Example

Problem

Simplify: 4 + 12 + 3 + 4 – 8.

4 + 12 + 3 + 4 – 8

12 + 3 + 4 + 4 + (−8)

12 + 3 + (4 + 4 + (−8))

12 + 3 + 0

12 + 3 + 0 = 15

Original expression.

Identify compatible numbers. 4 + 4 is 8, and tright here is a −8 existing. Recall that you have the right to think of – 8 as + (−8). Use the commutative residential property of enhancement to team them together.

Use the associative home to team 4 + 4 + (−8).

Add 4 + 4 + (−8).

Add the rest of the terms.

Answer 4 + 12 + 3 + 4 – 8 = 15


Simplify the expression: −5 + 25 – 15 + 2 + 8

A) 5

B) 15

C) 30

D) 55


Show/Hide Answer

A) 5

Incorrect. When you use the commutative home to rearselection the addends, make sure that negative addends lug their negative indications. The correct answer is 15.

B) 15

Correct. Use the commutative building to rearrange the expression so that compatible numbers are next to each various other, and then use the associative residential or commercial property to group them.

C) 30

Incorrect. Check your addition and also subtraction, and also think about the order in which you are adding these numbers. Use the commutative home to rearvariety the addends so that compatible numbers are next to each other. The correct answer is 15.

D) 55

Incorrect. It looks prefer you ignored the negative signs right here. When you use the commutative home to rearvariety the addends, make sure that negative addends bring their negative indicators. The correct answer is 15.

The Distributive Property


The distributive residential property of multiplication is an extremely helpful property that allows you recreate expressions in which you are multiplying a number by a amount or distinction. The property says that the product of a sum or distinction, such as 6(5 – 2), is equal to the sum or distinction of assets, in this situation, 6(5) – 6(2).

6(5 – 2) = 6(3) = 18

6(5) – 6(2) = 30 – 12 = 18

The distributive property of multiplication deserve to be provided as soon as you multiply a number by a sum. For instance, suppose you want to multiply 3 by the amount of 10 + 2.

3(10 + 2) = ?

According to this residential property, you can add the numbers 10 and also 2 first and also then multiply by 3, as presented here: 3(10 + 2) = 3(12) = 36. Alternatively, you can first multiply each addfinish by the 3 (this is referred to as distributing the 3), and then you can add the assets. This procedure is presented right here.

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3 (10 + 2) = 3(12) = 36

3(10) + 3(2) = 30 + 6 = 36

The products are the very same.

Since multiplication is commutative, you can usage the distributive property regardmuch less of the order of the components.

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The Distributive Properties

For any kind of genuine numbers a, b, and c:

Multiplication distributes over addition: a(b + c) = ab + ac

Multiplication distributes over subtraction: a(b – c) = ab – ac

Rewrite the expression 10(9 – 6) utilizing the distributive building.

A) 10(6) – 10(9)

B) 10(3)

C) 10(6 – 9)

D) 10(9) – 10(6)


Show/Hide Answer

A) 10(6) – 10(9)

Incorrect. Since subtraction isn’t commutative, you can’t change the order. The correct answer is 10(9) – 10(6).

B) 10(3)

Incorrect. This is a correct method to find the answer. But the question asked you to rewrite the difficulty making use of the distributive residential or commercial property. The correct answer is 10(9) – 10(6).

C) 10(6 – 9)

Incorrect. You readjusted the order of the 6 and the 9. Keep in mind that subtraction is not commutative and you did not usage the distributive home. The correct answer is 10(9) – 10(6).

D) 10(9) – 10(6)

Correct. The 10 is correctly distributed so that it is supplied to multiply the 9 and also the 6 separately.

Distributing via Variables


As lengthy as variables represent genuine numbers, the distributive building can be supplied via variables. The distributive property is crucial in algebra, and you will frequently view expressions like this: 3(x – 5). If you are asked to expand this expression, you can apply the distributive residential or commercial property just as you would if you were working through integers.

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3 (x – 5 ) = 3(x) – 3(5) = 3x – 15

Remember, once you multiply a number and a variable, you have the right to just write them side by side to express the multiplied quantity. So, the expression “three times the variable x” have the right to be composed in a variety of ways: 3x, 3(x), or 3 · x.


Example

Problem

Use the distributive home to expand the expression 9(4 + x).

9(4 + x)

9(4) + 9(x)

36 + 9x

Initial expression.

Distribute the 9 and multiply.

Multiply.

Answer 9(4 + x) = 36 + 9x


Example

Problem

Use the distributive residential property to evaluate the expression 5(2x – 3) once x = 2.

5(2x – 3)

5(2x) – 5(3)

10x – 15

10(2) – 15

20 – 15 = 5

Original expression.

Distribute the 5.

Multiply.

Substitute 2 for x, and evaluate.

Answer When x = 2, 5(2x – 3) = 5.


In the instance over, what perform you think would occur if you substituted x = 2 before distributing the 5? Would you acquire the same answer of 5? The instance below shows what would certainly happen.


Example

Problem

Use the distributive residential or commercial property to evaluate the expression 5(2x – 3) as soon as x = 2.

5(2x – 3)

5(2(2) – 3)

5(4 – 3)

5(4) – 5(3)

20 – 15 = 5

Initial expression.

Substitute 2 for x.

Multiply.

Subtract and also evaluate.

Answer When x = 2, 5(2x – 3) = 5.


Combining Like Terms


The distributive building have the right to additionally assist you understand a fundamental concept in algebra: that quantities such as 3x and 12x have the right to be added and subtracted in the very same method as the numbers 3 and 12. Let’s look at one instance and also watch exactly how it can be done.


Example

Problem

Add: 3x + 12x.

3(x) + 12(x)

x(3 + 12)

x(15)

or

15x

3x is 3 times x, and also 12x is 12 times x.

From studying the distributive residential property (and also using the commutative property), you understand that x(3 + 12) is the exact same as

3(x) + 12(x).

Combine the terms within the parentheses:

3 + 12 = 15.

Answer 3x + 12x = 15x


Do you see what happened? By thinking of the x as a dispersed quantity, you have the right to view that 3x + 12x = 15x. (If you’re not certain about this, attempt substituting any type of number for x in this expression…you will certainly discover that it holds true!)

Groups of terms that consist of a coefficient multiplied by the very same variable are referred to as “prefer terms”. The table listed below reflects some different teams of like terms:


Groups of Like Terms

3x, 7x, −8x, −0.5x

−1.1y, −4y, −8y

12t, 25t, 100t, 1t

4ab, −8ab, 2ab


Whenever you check out favor terms in an algebraic expression or equation, you deserve to add or subtract them just choose you would include or subtract real numbers. So, for instance,

10y + 12y = 22y, and 8x – 3x – 2x = 3x.

Be cautious not to incorporate terms that execute not have actually the same variable: 4x + 2y is not 6xy!


Example

Problem

Simplify: 10y + 5y + 9x – 6x – x.

10y + 5y + 9x – 6x – x

15y +  2x

Tright here are prefer terms in this expression, given that they all consist of a coeffective multiplied by the variable x or y. Keep in mind that – x is the very same as (−1)x.

Add choose terms. 10y + 5y = 15y, and

9x – 6x – x = 2x.

Answer 10y + 5y + 9x – 6x – x = 15y + 2x


Simplify: 12x – x + 2x – 8x.

A) 23x

B) 5

C) 5x

D) x


Show/Hide Answer

A) 23x

Incorrect. It looks like you added every one of the terms. Notice that −x and also −8x are negative. The correct answer is 5x.

B) 5

Incorrect. You unified the integers appropriately, but remember to encompass the variable too! The correct answer is 5x.

C) 5x

Correct. When you combine these like terms, you finish up via a amount of 5x.

D) x

Incorrect. It looks prefer you subtracted every one of the terms from 12x. Notice that −x and −8x are negative, but that 2x is positive. The correct answer is 5x.

See more: Why Is Carbon Important For Forming Complicated Molecules ? Or Why Carbon Makes A Large Number Of Molecules


Summary


The commutative, associative, and also distributive properties help you recreate a complicated algebraic expression into one that is much easier to resolve. When you recreate an expression by a commutative building, you adjust the order of the numbers being included or multiplied. When you recreate an expression utilizing an associative building, you group a different pair of numbers together using parentheses. You deserve to usage the commutative and also associative properties to regroup and also reorder any number in an expression as long as the expression is consisted of totally of addends or components (and also not a mix of them). The distributive building can be supplied to recreate expressions for a selection of purposes. When you are multiplying a number by a amount, you have the right to include and also then multiply. You deserve to also multiply each addfinish first and also then include the assets together. The exact same principle applies if you are multiplying a number by a distinction.