Asked by bluesanta
Which of the following correctly demonstrates the use of the Commutative Property of Multiplication?(1 point)
Responses
11(b+z)=11b+11z
11 left parenthesis b plus z right parenthesis equals 11 b plus 11 z
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
z11⋅2=z11+2
z superscript 11 baseline times 2 equals z superscript 11 baseline plus 2
3(b10+4)=3(4+b10)3
Responses
11(b+z)=11b+11z
11 left parenthesis b plus z right parenthesis equals 11 b plus 11 z
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
z11⋅2=z11+2
z superscript 11 baseline times 2 equals z superscript 11 baseline plus 2
3(b10+4)=3(4+b10)3
Answers
Answered by
bluesanta
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
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
Answered by
bluesanta
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
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
10+12a+a=10+13a
d plus 10 plus a equals 10 plus a plus d
12a+10=10+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
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
10+12a+a=10+13a
d plus 10 plus a equals 10 plus a plus d
12a+10=10+12a
Answered by
bluesanta
Substituting 1 for x in the equation 5x + 3=x⋅5 + 3 is a test case for which property?(1 point)
Responses
The Commutative Property of Addition
The Commutative Property of Addition
The Associative Property of Multiplication
The Associative Property of Multiplication
The Commutative Property of Multiplication
The Commutative Property of Multiplication
The Associative Property of Addition
Responses
The Commutative Property of Addition
The Commutative Property of Addition
The Associative Property of Multiplication
The Associative Property of Multiplication
The Commutative Property of Multiplication
The Commutative Property of Multiplication
The Associative Property of Addition
Answered by
bluesanta
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
No mistake was made.
No mistake was made.
Step 2
Step 2
Step 3
Step 3
Step 1
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
No mistake was made.
No mistake was made.
Step 2
Step 2
Step 3
Step 3
Step 1
Answered by
GPT-4o mini
The Commutative Property of Multiplication states that changing the order of the factors does not change the product. In other words, \( a \cdot b = b \cdot a \).
Let's analyze the given options:
1. **Option 1:** \( 11(b + z) = 11b + 11z \)
- This demonstrates the Distributive Property, not the Commutative Property.
2. **Option 2:** \( 2(b^{10} + z^{11}) = (b^{10} + z^{11}) \cdot 2 \)
- This shows that the order of multiplication does not affect the product, which is an example of the Commutative Property.
3. **Option 3:** \( z^{11} \cdot 2 = z^{11} + 2 \)
- This is incorrect as it does not hold true mathematically; it shows an incorrect operation.
4. **Option 4:** \( 3(b^{10} + 4) = 3(4 + b^{10}) \)
- This demonstrates that changing the order in which you add \( b^{10} \) and \( 4 \) does not change the sum (which is the Associative Property of Addition), along with the Commutative property of multiplication (since you can rewrite it as \( 3 \cdot (4 + b^{10}) \)).
From the above explanations, **Option 2** \( 2(b^{10} + z^{11}) = (b^{10} + z^{11}) \cdot 2 \) is the best example demonstrating the Commutative Property of Multiplication.
Let's analyze the given options:
1. **Option 1:** \( 11(b + z) = 11b + 11z \)
- This demonstrates the Distributive Property, not the Commutative Property.
2. **Option 2:** \( 2(b^{10} + z^{11}) = (b^{10} + z^{11}) \cdot 2 \)
- This shows that the order of multiplication does not affect the product, which is an example of the Commutative Property.
3. **Option 3:** \( z^{11} \cdot 2 = z^{11} + 2 \)
- This is incorrect as it does not hold true mathematically; it shows an incorrect operation.
4. **Option 4:** \( 3(b^{10} + 4) = 3(4 + b^{10}) \)
- This demonstrates that changing the order in which you add \( b^{10} \) and \( 4 \) does not change the sum (which is the Associative Property of Addition), along with the Commutative property of multiplication (since you can rewrite it as \( 3 \cdot (4 + b^{10}) \)).
From the above explanations, **Option 2** \( 2(b^{10} + z^{11}) = (b^{10} + z^{11}) \cdot 2 \) is the best example demonstrating the Commutative Property of Multiplication.
Answered by
GPT-4o mini
The Commutative Property of Addition states that changing the order of the addends does not change the sum. In other words, \( a + b = b + a \).
Let's analyze the given options:
1. **Option 1:** \( abcxyz = zyxcba \)
- This does not pertain to addition; it appears to be a rearrangement of characters rather than a mathematical expression demonstrating the Commutative Property.
2. **Option 2:** \( 123 + 456 = 579 \)
- This mathematically shows an equation, but it does not demonstrate the Commutative Property since it does not illustrate exchanging the order of the addends.
3. **Option 3:** \( 0 + 579 = 579 \)
- This indicates the identity element in addition, but it does not demonstrate the Commutative Property.
4. **Option 4:** \( abc + xyz = xyz + abc \)
- This correctly illustrates the Commutative Property of Addition, as it shows that changing the order of the addends (from \(abc + xyz\) to \(xyz + abc\)) does not change the sum.
Therefore, the correct response that demonstrates the Commutative Property of Addition is **Option 4: \( abc + xyz = xyz + abc \)**.
Let's analyze the given options:
1. **Option 1:** \( abcxyz = zyxcba \)
- This does not pertain to addition; it appears to be a rearrangement of characters rather than a mathematical expression demonstrating the Commutative Property.
2. **Option 2:** \( 123 + 456 = 579 \)
- This mathematically shows an equation, but it does not demonstrate the Commutative Property since it does not illustrate exchanging the order of the addends.
3. **Option 3:** \( 0 + 579 = 579 \)
- This indicates the identity element in addition, but it does not demonstrate the Commutative Property.
4. **Option 4:** \( abc + xyz = xyz + abc \)
- This correctly illustrates the Commutative Property of Addition, as it shows that changing the order of the addends (from \(abc + xyz\) to \(xyz + abc\)) does not change the sum.
Therefore, the correct response that demonstrates the Commutative Property of Addition is **Option 4: \( abc + xyz = xyz + abc \)**.
Answered by
GPT-4o mini
To use the Commutative Property to determine the missing step in proving the equivalence of \( 12a + 10 + a = 10 + a + 12a \), we need to manipulate the left side to reorder the terms appropriately.
**Step 1** should demonstrate rearranging the terms in the expression \( 12a + 10 + a \) to match the right side of the equation.
From the original left side:
- You can group \( 12a \) and \( a \) to make it simpler to apply the Commutative Property.
The correct transformation for **Step 1** is:
- **\( 12a + 10 + a = 10 + a + 12a \)**
This means we can see that adding \( a \) after \( 10 \) and before \( 12a \) follows the Commutative Property of Addition.
Looking at the given options:
1. **\( 12a + 10 + a - 10 = 10 + a + 12a - 10 \)**
- This introduces a subtraction (which is not related to the Commutative Property) and is incorrect.
2. **\( 12a + 10 + a - a = 10 + a + 12a - a \)**
- This again introduces subtraction and is not applicable for this context.
3. **\( 10 + 12a + a = 10 + 13a \)**
- This does not present proper reordering for the first step; it misrepresents the left expression.
4. **\( 12a + 10 = 10 + 12a \)**
- This option does not include the \( a \) term, which is necessary for proving the equivalence.
None of the options accurately capture the use of the Commutative Property as the missing step should show rearranging the terms directly without introducing unnecessary elements like subtraction or incorrectly omitting terms.
However, based on what we know, we can state that the intent of **Step 1** should effectively be something like:
**12a + 10 + a = 10 + a + 12a** (not matching an option provided).
Given the responses, they don't match exactly but the closest conceptually would be distinct from the provided answers.
Ultimately, the missing step, correctly stated, should highlight:
**Step 1:** Rearranging gives us \( 10 + 12a + a = 10 + a + 12a \), applying the arithmetic rearrangement successfully showcases the property.
**Step 1** should demonstrate rearranging the terms in the expression \( 12a + 10 + a \) to match the right side of the equation.
From the original left side:
- You can group \( 12a \) and \( a \) to make it simpler to apply the Commutative Property.
The correct transformation for **Step 1** is:
- **\( 12a + 10 + a = 10 + a + 12a \)**
This means we can see that adding \( a \) after \( 10 \) and before \( 12a \) follows the Commutative Property of Addition.
Looking at the given options:
1. **\( 12a + 10 + a - 10 = 10 + a + 12a - 10 \)**
- This introduces a subtraction (which is not related to the Commutative Property) and is incorrect.
2. **\( 12a + 10 + a - a = 10 + a + 12a - a \)**
- This again introduces subtraction and is not applicable for this context.
3. **\( 10 + 12a + a = 10 + 13a \)**
- This does not present proper reordering for the first step; it misrepresents the left expression.
4. **\( 12a + 10 = 10 + 12a \)**
- This option does not include the \( a \) term, which is necessary for proving the equivalence.
None of the options accurately capture the use of the Commutative Property as the missing step should show rearranging the terms directly without introducing unnecessary elements like subtraction or incorrectly omitting terms.
However, based on what we know, we can state that the intent of **Step 1** should effectively be something like:
**12a + 10 + a = 10 + a + 12a** (not matching an option provided).
Given the responses, they don't match exactly but the closest conceptually would be distinct from the provided answers.
Ultimately, the missing step, correctly stated, should highlight:
**Step 1:** Rearranging gives us \( 10 + 12a + a = 10 + a + 12a \), applying the arithmetic rearrangement successfully showcases the property.
Answered by
GPT-4o mini
To determine which property is being tested by substituting \( x = 1 \) in the equation \( 5x + 3 = x \cdot 5 + 3 \), we need to analyze the equation and the properties involved.
1. **The Commutative Property of Addition** states that changing the order in which two numbers are added does not change the sum (e.g., \( a + b = b + a \)).
2. **The Associative Property of Multiplication** states that the way in which factors are grouped in a multiplication expression does not change the product (e.g., \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)).
3. **The Commutative Property of Multiplication** states that changing the order of the factors does not change the product (e.g., \( a \cdot b = b \cdot a \)).
4. **The Associative Property of Addition** states that the way in which numbers are grouped in an addition expression does not change the sum (e.g., \( (a + b) + c = a + (b + c) \)).
In the equation \( 5x + 3 = x \cdot 5 + 3 \):
- The left side can be seen as \( 5x + 3 \).
- The right side can be rearranged using the Commutative Property of Multiplication because you are effectively rewriting \( x \cdot 5 \) as \( 5 \cdot x \).
By substituting \( x = 1 \), you are testing whether both sides of the equation yield the same result. This is also reflected in the equation's development involving multiplication.
Thus, the substitution of \( 1 \) for \( x \) effectively highlights the application of the **Commutative Property of Multiplication** in the context of the expression.
Therefore, the correct response is:
**The Commutative Property of Multiplication.**
1. **The Commutative Property of Addition** states that changing the order in which two numbers are added does not change the sum (e.g., \( a + b = b + a \)).
2. **The Associative Property of Multiplication** states that the way in which factors are grouped in a multiplication expression does not change the product (e.g., \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)).
3. **The Commutative Property of Multiplication** states that changing the order of the factors does not change the product (e.g., \( a \cdot b = b \cdot a \)).
4. **The Associative Property of Addition** states that the way in which numbers are grouped in an addition expression does not change the sum (e.g., \( (a + b) + c = a + (b + c) \)).
In the equation \( 5x + 3 = x \cdot 5 + 3 \):
- The left side can be seen as \( 5x + 3 \).
- The right side can be rearranged using the Commutative Property of Multiplication because you are effectively rewriting \( x \cdot 5 \) as \( 5 \cdot x \).
By substituting \( x = 1 \), you are testing whether both sides of the equation yield the same result. This is also reflected in the equation's development involving multiplication.
Thus, the substitution of \( 1 \) for \( x \) effectively highlights the application of the **Commutative Property of Multiplication** in the context of the expression.
Therefore, the correct response is:
**The Commutative Property of Multiplication.**
Answered by
GPT-4o mini
To identify any errors in the steps provided while using the Commutative Property of Addition to verify the equation \( 8 + x^2 + 17x - x^2 + 4 = 10 + 4x^2 + 10x - 4x^2 + 2 + 7x \), let's break down each step carefully.
### Step 1
\[ 8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \]
- This step appears to be correctly rearranging terms using the Commutative Property of Addition. All terms are present in some order.
### Step 2
\[ 8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \]
- Here, the left side correctly combines like terms, as \( x^2 - x^2 \) will cancel, giving:
\[ 8 + 4 + 17x = 12 + 17x \]
- The right side also combines correctly, as \( 10 + 2 + (4x^2 - 4x^2) + 10x + 7x = 12 + 17x \).
### Step 3
\[ 12 + 17x + x^2 = 12 - x^2 + 17x \]
- In this step, there is an inconsistency. The left side had \( 12 + 17x + x^2 \), but the right side has \( 12 - x^2 + 17x \). The left-hand side should logically have canceled \( x^2 \) entirely, resulting in no term for \( x^2 \).
### Conclusion
Thus, the mistake occurs in **Step 3**, where \( + x^2 \) is incorrectly reflected as \( - x^2 \) on the right side.
Therefore, the correct response indicating where the mistake was made is:
**Step 3**.
### Step 1
\[ 8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \]
- This step appears to be correctly rearranging terms using the Commutative Property of Addition. All terms are present in some order.
### Step 2
\[ 8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \]
- Here, the left side correctly combines like terms, as \( x^2 - x^2 \) will cancel, giving:
\[ 8 + 4 + 17x = 12 + 17x \]
- The right side also combines correctly, as \( 10 + 2 + (4x^2 - 4x^2) + 10x + 7x = 12 + 17x \).
### Step 3
\[ 12 + 17x + x^2 = 12 - x^2 + 17x \]
- In this step, there is an inconsistency. The left side had \( 12 + 17x + x^2 \), but the right side has \( 12 - x^2 + 17x \). The left-hand side should logically have canceled \( x^2 \) entirely, resulting in no term for \( x^2 \).
### Conclusion
Thus, the mistake occurs in **Step 3**, where \( + x^2 \) is incorrectly reflected as \( - x^2 \) on the right side.
Therefore, the correct response indicating where the mistake was made is:
**Step 3**.
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