Which of the following is an equation with one constant, two coefficients, and three terms?(1 point)

Question

Responses

3x−y=5y
3 x minus y equals 5 y

3x−4y=5y
3 x minus 4 y equals 5 y

3x−4y=5x
3 x minus 4 y equals 5 x

3x−4=5y

Answer 1

An equation with one constant, two coefficients, and three terms will have the following structure:

• One constant term (a standalone number).
• Two coefficient terms (terms with variables that have numerical coefficients).
• Three total terms.

Looking at the options provided:

1. 3x−y=5y → This simplifies to 3x - 6y = 0, which would have only two terms in the form of a polynomial (3x and -6y).
2. 3x−4y=5y → This can be rearranged to 3x - 9y = 0. This has three terms (3x, -9y, and 0 which can be considered the constant term implicitly).
3. 3x−4y=5x → Rearranging gives -2x - 4y = 0, again having two variables.
4. 3x−4=5y → This rearranges to 3x - 5y - 4 = 0 and has three terms (3x, -5y, and -4).

Both 3x−4y=5y and 3x−4=5y can be argued to have three terms when rewritten in standard form.

However, the option 3x−4=5y clearly presents just one constant term (-4), two coefficient terms (3x and 5y), and a total of three unique terms.

Thus, the answer is:

3x−4=5y.

Answer 2

Consider the expression x+632−(2⋅3)+5
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6
3
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2
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5
. What is the simplified denominator?(1 point)
Responses

−2
−
2
negative 2

56
56

8
8

26

Answer 3

Let's simplify the expression step by step:

The expression given is:

\[ x + 632 - (2 \cdot 3) + 5 \]

1. Calculate \( (2 \cdot 3) \): \[ 2 \cdot 3 = 6 \]

2. Substitute back into the expression: \[ x + 632 - 6 + 5 \]

3. Combine the constant terms: \[ 632 - 6 + 5 = 632 - 1 = 631 \]

So, the expression simplifies to:

\[ x + 631 \]

As for determining the simplified denominator in the traditional sense, since there is no fraction provided in the expression, this does not seem to fit the usual context of a "denominator." However, if we consider just the constants involved:

If you want to think of a form where you have a denominator as in \( \frac{numerator}{denominator} \), then clearly, there is no implied fraction here to generate a simplified denominator from the expression.

Therefore, if this focuses solely on constants and doesn't lead to an actual denominator (when not framed as a fraction), there isn't a "simplified denominator." If you meant something else, please clarify!

In the context you provided, out of the options given, there is no clear denominator derived from the given expression. If it must adhere to one option, it would most likely relate back to another context not completely defined here.

If more context is available, I would be glad to assist accordingly.

Answer 4

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

5z9+8a=8a+5z9
5
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8
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8
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5
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5 z superscript 9 baseline plus 8 a equals 8 a plus 5 z superscript 9 baseline

5+8=4+9
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4
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9
5 plus 8 equals 4 plus 9

0+5z9=5z9
0
+
5
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9
=
5
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0 plus 5 z superscript 9 baseline equals 5 z superscript 9 baseline

z9⋅5=5z9
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9
⋅
5
=
5
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9
z superscript 9 baseline times 5 equals 5 z superscript 9 baseline

Answer 5

The Commutative Property of Addition states that the order in which numbers are added does not change the sum. In other words, for any two numbers \(a\) and \(b\):

\[ a + b = b + a \]

Now let’s evaluate each of the given responses to determine which one correctly demonstrates the Commutative Property of Addition:

1. \(5z^9 + 8a = 8a + 5z^9\)
   This shows that the order of addition is switched, thus it demonstrates the Commutative Property.

2. \(5 + 8 = 4 + 9\)
   This is an equality statement, not an example of the Commutative Property, as it doesn't show the same numbers being added in a different order.

3. \(0 + 5z^9 = 5z^9\)
   This is a case of the Identity Property of Addition, where adding zero does not change the value of \(5z^9\), but it does not demonstrate the Commutative Property.

4. \(z^9 \cdot 5 = 5z^9\)
   This is an example of the Commutative Property of Multiplication, not addition.

The correct choice that demonstrates the Commutative Property of Addition is:

\(5z^9 + 8a = 8a + 5z^9\).

Answer 6

Which expressions correctly demonstrate the Commutative Property?(1 point)
Responses

3x⋅9y⋅6=9y⋅6⋅3x
3
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9
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6
=
9
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6
⋅
3
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3 x times 9 y times 6 equals 9 y times 6 times 3 x

3x⋅9y⋅6=162xy
3
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9
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6
=
162
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3 x times 9 y times 6 equals 162 x y

3x⋅9y⋅6=9y+6+3x
3
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9
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⋅
6
=
9
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6
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3
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3 x times 9 y times 6 equals 9 y plus 6 plus 3 x

3x(9y+6)=(3x+9y)⋅6

Answer 7

The Commutative Property states that the order of operations does not affect the result. This property applies to both addition and multiplication. Specifically, for multiplication, it states that for any two factors \(a\) and \(b\):

\[ a \cdot b = b \cdot a \]

Now, let's evaluate each of the provided expressions to see which ones correctly demonstrate the Commutative Property:

1. \(3x \cdot 9y \cdot 6 = 9y \cdot 6 \cdot 3x\)
   This expression shows that the order of multiplication is switched, thus demonstrating the Commutative Property.

2. \(3x \cdot 9y \cdot 6 = 162xy\)
   This shows a multiplication operation resulting in a product, but it does not demonstrate the Commutative Property, as it does not show the order being changed.

3. \(3x \cdot 9y \cdot 6 = 9y + 6 + 3x\)
   This expression incorrectly mixes multiplication and addition and does not demonstrate the Commutative Property. The left side represents a product, while the right side represents a sum.

4. \(3x(9y + 6) = (3x + 9y) \cdot 6\)
   This expression illustrates the distributive property, not the commutative property. It does not demonstrate the property of changing the order of multiplication.

The only expression that correctly demonstrates the Commutative Property is:

\(3x \cdot 9y \cdot 6 = 9y \cdot 6 \cdot 3x\).

Answer 8

According to the Associative Property of Addition, which of the following is equivalent to the expression (b+c)+a+d
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?(1 point)
Responses

d+(b+c)+a
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d plus left parenthesis b plus c right parenthesis plus a

(b+c)+d+a
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left parenthesis b plus c right parenthesis plus d plus a

b+(c+a)+d
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b plus left parenthesis c plus a right parenthesis plus d

d+a+(b+c)
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)

Answer 9

The Associative Property of Addition states that when adding three or more numbers, the way in which the numbers are grouped does not change the sum. Mathematically, for any numbers \(a\), \(b\), and \(c\):

\[ (a + b) + c = a + (b + c) \]

Given the expression \((b + c) + a + d\), we want to find an equivalent expression that groups the additions in a different way while maintaining the same total.

Let's evaluate each of the provided options:

1. \(d + (b + c) + a\)
   This is equivalent, as the grouping is changed but all terms are included and can be regrouped. This uses the Associative Property.

2. \((b + c) + d + a\)
   This keeps the original grouping and, as such, is equivalent to \((b + c) + a + d\).

3. \(b + (c + a) + d\)
   This reorders the grouping of \(c\) and \(a\), but it also results in the same total, which makes it equivalent.

4. \(d + a + (b + c)\)
   This expression is also valid and demonstrates the Associative Property, where the terms can be rearranged.

All options produced valid combinations resulting from the Associative Property.

However, since the goal is to derive equivalency, the most notable transformations based purely on the initial grouping would be:

\(d + (b + c) + a\) and \((b + c) + d + a\) would remain the same structurally, preserving the initial expression's components. But so would \(b + (c + a) + d\) and \(d + a + (b + c)\) considering full reassessments of the individual variables involved.

All four options are mathematically equivalent, but if you were to choose something that distinctively illustrated just an alternate group from: \((b + c) + a + d\), the emphasis would lean towards the top choice:

\(d + (b + c) + a\).

Answer 10

wrong answer

Answer 11

I apologize for any confusion. Let's clarify the Associative Property of Addition in relation to the specific expression given: \((b + c) + a + d\).

The Associative Property states that the grouping of the numbers does not affect their sum. Therefore, we can rearrange the parentheses in the expression, but we cannot change the order of the numbers.

Revisiting your options:

1. \(d + (b + c) + a\)
   This correctly shows that \(d\) is grouped separately, which is valid.

2. \((b + c) + d + a\)
   This simply shifts the grouping of \(d + a\), maintaining the same groups (still equivalent).

3. \(b + (c + a) + d\)
   This expression changes the grouping of \(c\) and \(a\), which does not respect the original grouping of \((b + c)\) and introduces a different structure—this is not equivalent.

4. \(d + a + (b + c)\)
   This expression also shows an alteration in the grouping of terms, not preserving initial structure—though still valid in its sum.

Thus, the expressions that respect the original structure of the addition using the associative property would be the first and second options.

Given this, the correct expressions that are equivalent to the provided expression \((b + c) + a + d\) while using the Associative Property would ultimately be:

1. \(d + (b + c) + a\)
2. \((b + c) + d + a\)
3. \(d + a + (b + c)\) can be viewed, but it shifts from our original base.

The most straightforward and direct answer that maintains the integrity of \( (b + c) \) would indeed be \(d + (b + c) + a\), which maintains all terms while altering only the grouping.

Thank you for your patience, and I hope this clarifies the application of the Associative Property accurately!