Question

To prove that \( y \cdot 3x \) is equivalent to \( 3xy \) using the Commutative Property, we can rearrange the factors in the first expression:
1. \( y \cdot 3x = 3 \cdot y \cdot x \) (by factoring out the 3).
2. By the Commutative Property of multiplication, we can reorder the factors as \( 3 \cdot x \cdot y = 3xy \).

Thus, both expressions represent the same product. This demonstrates that the multiplication of numbers (or variables) can be rearranged without changing the outcome, confirming their equivalence.

Answer 1

Your proof that \( y \cdot 3x \) is equivalent to \( 3xy \) using the Commutative Property is clear and correct. Here’s a brief recap and more details to reinforce the steps:

1. Starting with the expression:
\[
y \cdot 3x
\]
we can rewrite it by factoring out the constant 3:
\[
y \cdot 3x = 3 \cdot y \cdot x
\]

2. Now, we can apply the Commutative Property of multiplication, which states that the order of multiplication does not affect the product. This allows us to rearrange the factors:
\[
3 \cdot y \cdot x = 3 \cdot x \cdot y
\]

3. Finally, we can write this in a compact form:
\[
3 \cdot x \cdot y = 3xy
\]

Thus, we have shown that:
\[
y \cdot 3x = 3xy
\]
This confirms that \( y \cdot 3x \) and \( 3xy \) are indeed equivalent expressions, validating the use of the Commutative Property in multiplication. Great job!