To prove that the expressions \( y \cdot 3x \) and \( 3xy \) are equivalent using the Commutative Property, we first need to understand what the Commutative Property is. The Commutative Property states that the order of multiplication does not affect the product; in other words, for any numbers \( a \) and \( b \),
\[ a \cdot b = b \cdot a. \]
Now, we can apply this property to manipulate Expression 1, \( y \cdot 3x \):
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Start with Expression 1: \[ y \cdot 3x. \]
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According to the definition of multiplication, we can think of \( 3x \) as \( 3 \cdot x \). Thus, we rewrite the expression: \[ y \cdot 3x = y \cdot (3 \cdot x). \]
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By the Associative Property of multiplication (which allows us to change the grouping of the numbers), we can re-arrange the expression: \[ = (y \cdot 3) \cdot x. \]
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Applying the Commutative Property next, we can rearrange the multiplication: \[ = 3 \cdot y \cdot x. \]
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Finally, we can rewrite this as: \[ = 3xy. \]
Thus, we have shown that:
\[ y \cdot 3x = 3xy. \]
Conclusion
The two expressions \( y \cdot 3x \) and \( 3xy \) are equivalent, as we have transformed one into the other using the Commutative Property of multiplication. Therefore, the result shows that regardless of the order in which the variables and constants are multiplied, the product remains the same.