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Which of the following correctly demonstrates the use of the Commutative Property of Multiplication?(1 point)

Responses

z11⋅2=z11+2
z superscript 11 baseline times 2 equals z superscript 11 baseline plus 2

2(b10+z11)=(b10+z11)⋅2
2 left parenthesis b superscript 10 baseline plus z superscript 11 baseline right parenthesis equals left parenthesis b superscript 10 baseline plus z superscript 11 baseline right parenthesis times 2

3(b10+4)=3(4+b10)3
3 left parenthesis b superscript 10 baseline plus 4 right parenthesis equals 3 left parenthesis 4 plus b superscript 10 baseline right parenthesis 3

11(b+z)=11b+11z

9 answers

The Commutative Property of Multiplication states that changing the order of the factors does not change the product. In other words, \( a \cdot b = b \cdot a \).

The correct demonstration of the Commutative Property of Multiplication among the given responses would be:

3(b10 + 4) = 3(4 + b10)

This shows that the order of 4 and b10 in the sum does not affect the product with 3, which is in line with the commutative property of addition. However, if you are looking specifically for an example of just multiplication without the context of addition, the closest related example might be found in the expression, but keep in mind the original property strictly refers to the multiplication of two numbers.

If you're strictly looking for the multiplication aspect, none of the options fully represent the classic Commutative Property of Multiplication directly; instead, they demonstrate properties of addition or distribution. The closest in form is the one that rearranges the addition inside the parentheses.

If the question allows for the interpretation involving the addition format or contexts, then 3(b10 + 4) = 3(4 + b10) could be deemed acceptable in showing the reordering characteristic consistent with the properties of arithmetic operations but not directly the multiplication property since it's involving summands.

If you must choose strictly the best option regarding the Commutative Property, this is the response to consider.

Which of the following correctly demonstrates the Commutative Property of Addition?(1 point)
Responses

abcxyz=zyxcba
a b c x y z equals z y x c b a

123+456=579
123 plus 456 equals 579

0+579=579
0 plus 579 equals 579

abc+xyz=xyz+abc

The Commutative Property of Addition states that changing the order of the addends does not change the sum. In mathematical terms, this can be expressed as:

\[ a + b = b + a \]

Among the given responses, the one that correctly demonstrates the Commutative Property of Addition is:

abc + xyz = xyz + abc

This expression shows that the order of the terms (abc and xyz) can be switched without affecting the sum, which is the essence of the Commutative Property of Addition.

Use the Commutative Property to determine the missing step in proving the equivalence of 12a+10+a=10+a+12a.

Step 1: [missing]

Step 2: 10+13a=10+13a

Step 3: The expressions are equivalent because they both equal the same value.

(1 point)
Responses

10+12a+a=10+13a
d plus 10 plus a equals 10 plus a plus d

12a+10+a−10=10+a+12a−10
d plus 10 plus a equals 10 plus a plus d

12a+10+a−a=10+a+12a−a
d plus 10 plus a equals 10 plus a plus d

12a+10=10+12a

To find the missing step using the Commutative Property, we should focus on manipulating the left-hand side of the original equation \(12a + 10 + a\) to reflect the right-hand side \(10 + a + 12a\).

  1. The original expression is \(12a + 10 + a\).
  2. We can rearrange the terms using the Commutative Property of Addition, which allows us to change the order of the addends.

The correct choice for the missing step is:

12a + 10 + a = 10 + a + 12a

This step correctly demonstrates the Commutative Property of Addition by rearranging the terms in the expression.

After this step, you would then simplify to arrive at:

Step 2: 10 + 13a = 10 + 13a

And finally, conclude that the expressions are equivalent because they both equal the same value.

Substituting 1 for x in the equation 5x + 3=x⋅5 + 3 is a test case for which property?(1 point)
Responses

The Associative Property of Addition
The Associative Property of Addition

The Commutative Property of Addition
The Commutative Property of Addition

The Commutative Property of Multiplication
The Commutative Property of Multiplication

The Associative Property of Multiplication

Substituting 1 for \( x \) in the equation \( 5x + 3 = x \cdot 5 + 3 \) is demonstrating the validity of the equation based on the property of multiplication and addition but more specifically reflects the reordering of terms.

The equation actually shows:

\[ 5x = x \cdot 5 \]

This is an example of the Commutative Property of Multiplication, because it illustrates that multiplying \( x \) by 5 (i.e., \( x \cdot 5 \)) is the same as multiplying 5 by \( x \) (i.e., \( 5x \)).

So, the correct response is:

The Commutative Property of Multiplication.

Which step contains an error using the Commutative Property of Addition in verifying 8+x2+17x−x2+4=10+4x2+10x−4x2+2+7x?

Step 1: 8+4+x2−x2+17x=10+2+10x−4x2+7x +4x2

Step 2: 8+4+17x+x2−x2=10+2+4x2−4x2+10x+7x

Step 3: 12+17x+x2=12−x2+17x

(1 point)
Responses

Step 2
Step 2

Step 3
Step 3

Step 1
Step 1

No mistake was made.

To determine if there is an error in the verification process using the Commutative Property of Addition, we can review each step carefully.

Step 1:

\[ 8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \]

This step looks correct at first glance as it maintains equality and correct arrangement of terms.

Step 2:

\[ 8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \]

Again, this step seems to correctly rearrange terms considering the Commutative Property of Addition. However, check if both sides still balance.

Left side: \[ 8 + 4 = 12, \] \[ 12 + 17x + x^2 - x^2 = 12 + 17x \]

Right side: \[ 10 + 2 = 12, \] \[ 12 + 4x^2 - 4x^2 + 10x + 7x = 12 + 10x + 7x = 12 + 17x \]

Both sides still balance.

Step 3:

\[ 12 + 17x + x^2 = 12 - x^2 + 17x \]

In this step, there is an error because the left side \( 12 + 17x + x^2 \) does not equal the right side \( 12 - x^2 + 17x \).

The correct rearrangement should keep the \( x^2 \) term positive on both sides, not switch it to negative.

Therefore, the step that contains an error is:

Step 3.