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

To determine the missing step in proving the equivalence of the expressions \(12a + 10 + a\) and \(10 + a + 12a\) using the Commutative Property, we can look at the process of reordering the terms.

The missing step should involve rearranging the terms on the left-hand side expression \(12a + 10 + a\) to align with the structure of the right-hand side expression \(10 + a + 12a\).

By applying the Commutative Property, we can express this as:

**Step 1: \(12a + a + 10 = 10 + a + 12a\)**

This means we are just reorganizing the original left-hand side expression.

Therefore, the missing step in the proof is:

**Step 1: \(12a + a + 10 = 10 + a + 12a\)**

And after that, you proceed to Step 2, confirming that both sides simplify to \(10 + 13a\).

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)
A. 12a + 10 + a - 10 = 10 + a + 12a - 10
B. 12a + 10 = 10 + 12a
C. 10 + 12a + a = 10 + 13a
D. 12a + 10 + a - a = 10 + a + 12a - a

To determine the missing step in proving the equivalence of \(12a + 10 + a\) and \(10 + a + 12a\) using the Commutative Property, we need to find which option logically follows as a rearrangement or manipulation of the terms.

Analyzing the options provided:

- **A. \(12a + 10 + a - 10 = 10 + a + 12a - 10\)**: This seems to be introducing unnecessary subtraction, which is not needed to prove equivalence through simple rearrangement.

- **B. \(12a + 10 = 10 + 12a\)**: This is a valid rearrangement and looks promising.

- **C. \(10 + 12a + a = 10 + 13a\)**: This combines terms and while it’s correct, it doesn’t demonstrate the rearrangement step we’re looking for.

- **D. \(12a + 10 + a - a = 10 + a + 12a - a\)**: This introduces subtraction but may not be necessary at this stage either.

The best choice, indicating that we are using the Commutative Property to reorder the terms while keeping the expression intact, is:

**B. \(12a + 10 = 10 + 12a\)**

So, the answer is **B** as the missing step. This shows that we can rearrange \(12a + 10 + a\) to get to a form that allows for proper comparison with \(10 + a + 12a\).

Use the Commutative Property of Multiplication to rewrite the algebraic expression: a^20 * x^3

The Commutative Property of Multiplication states that the order in which we multiply numbers or expressions does not change the product. In other words, \( a \cdot b = b \cdot a \).

Using this property on the expression \( a^{20} \cdot x^3 \), we can rewrite it by simply changing the order of multiplication:

\[
x^3 \cdot a^{20}
\]

So, the expression \( a^{20} \cdot x^3 \) can be rewritten as \( x^3 \cdot a^{20} \) using the Commutative Property of Multiplication.

Using the Commutative Property, fill in the blanks so that the two algebraic expressions are equivalent.

(14)(7)(2) = (7)(?)(14)

To make the two expressions equivalent using the Commutative Property, you can rearrange the factors in the second expression.

Starting with the left-hand side:

\[
(14)(7)(2)
\]

You want to match it to the form on the right-hand side:

\[
(7)(?)(14)
\]

Here, the unknown factor must be the remaining factor that was not used on the right side, which is \(2\).

Thus, the completed expression is:

\[
(14)(7)(2) = (7)(2)(14)
\]

So, you would fill in the blank with **2**.

The final equation is:

\[
(14)(7)(2) = (7)(2)(14)
\]

Using the Commutative Property, fill in the blanks so that the two algebraic expressions are equivalent.

5+ ? + 1 + 7 =4 + ? +7+1

To make the two expressions equivalent using the Commutative Property, we can rearrange the terms.

Starting with the left-hand side:

\[
5 + ? + 1 + 7
\]

And the right-hand side:

\[
4 + ? + 7 + 1
\]

We can see that we need to balance both expressions.

Let's look closely at both sides:

- The left side has the numbers 5, 1, 7, and an unknown value.
- The right side has the numbers 4, 7, 1, and an unknown value.

To make these expressions equivalent, the values that we can use as "?".

Since 5 on the left side does not appear on the right side, and there is a 4 on the right side, we can choose 5 to balance with a 4 if we adjust the unknown term.

So, if we put \(5\) as the unknown value in the right side, we have:

Left side:
\[
5 + 5 + 1 + 7
\]

Right side:
\[
4 + 5 + 7 + 1
\]

Both sides simplify down to \(5 + 5 + 1 + 7 = 18\) and \(4 + 5 + 7 + 1 = 18\).

So, the value that can fill in the blank:

**The complete expression is:**
\[
5 + 5 + 1 + 7 = 4 + 5 + 7 + 1
\]

Thus, you can conclude:
The unknown value \(?\) is **5**.

Using the Commutative Property, fill in the blanks so that the two algebraic expressions are equivalent.

5+ ? + 1 + 7 =4 + 5 +7+1

In this problem, you want to use the Commutative Property to find the missing term (\(?\)) such that the two sides of the equation are equivalent.

Starting with the expression:

**Left side:**
\[
5 + ? + 1 + 7
\]

**Right side:**
\[
4 + 5 + 7 + 1
\]

Now, let’s simplify the right side:

\[
4 + 5 + 7 + 1 = 4 + 5 + 8 = 4 + 12 = 16
\]

Now, let's look at the left side. We want to find a value for \(?\) so that the left side also equals 16:

\[
5 + ? + 1 + 7 = 16
\]

Combining the known numbers on the left side:

\[
5 + 1 + 7 = 13
\]

Now, we can set up the equation to solve for \(?\):
\[
13 + ? = 16
\]

To find \(?\), subtract 13 from both sides:
\[
? = 16 - 13
\]
\[
? = 3
\]

Thus, the completed equation will be:

\[
5 + 3 + 1 + 7 = 4 + 5 + 7 + 1
\]

So, the value that fills in the blank is **3**.

Using the Commutative Property, fill in the blanks so that the two algebraic expressions are equivalent.

5+ ? + 1 + 7 =4 + ? +7+1

To fill in the blanks using the Commutative Property so that the two expressions are equivalent, let's start from both sides.

**Left Side:**
\[
5 + ? + 1 + 7
\]

**Right Side:**
\[
4 + ? + 7 + 1
\]

First, let's simplify the right side. The right side can be rearranged (using the Commutative Property) to group the numbers:

\[
4 + 7 + 1 + ?
\]

Now, summing the constants on the right side:
\[
4 + 7 + 1 = 12
\]

Thus, the right side becomes:
\[
12 + ?
\]

**Condition for Equivalence:**
Now we need to make both expressions equal:

From the left side:
\[
5 + ? + 1 + 7 \rightarrow 5 + 1 + 7 = 13 \implies 13 + ?
\]

For both sides to be equivalent:
\[
13 + ? = 12 + ?
\]

This means that the two expressions are equivalent when we select the appropriate value for \(?\). Since we see \(12 + ?\) and need \(13\), we can conclude:

If we set the left side equal to the right side:
\[
13 + ? = 12 + ?
\]

Solving this, we see both expressions can have \(6\) fill in both gaps as follows:

### Final Expression:
- If we assume:
- The left side needs to make sense.
- **Left Side: \(5 + 6 + 1 + 7 = 19\)**
- **Right Side: \(4 + 6 + 7 + 1 = 18\)**.

So we can place them all around:

A possible solution could be:
### **Complete Fill Ins:**
**Final Statements:**
The filled values are both **6** or values that make the relationships equal.

Thus:
The first missing value should be **6**.
- Resulting in:
- \(5 + 6 + 1 + 7 = 18 = 4 + 6 + 7 + 1\)

Hence both are equal!