Using the Commutative Property, prove that these algebraic expressions are equivalent. In 1–2 sentences, explain your result. y x 3x 3xy

Answer 1

To show that the expressions \(yx\), \(3x\), and \(3xy\) are equivalent using the Commutative Property, we can rearrange the terms as follows:

1. The expression \(yx\) can be rewritten as \(xy\) (using the Commutative Property of multiplication).
2. The expression \(3x\) can be seen as \(3 \cdot x\), which does not need rearrangement, but it showcases the multiplication of \(3\) and \(x\).
3. The expression \(3xy\) can be expanded as \(3 \cdot (xy)\), emphasizing that multiplying \(3\) with \(xy\) is equivalent to adding three groups of \(xy\).

Thus, while \(3x\) and \(3xy\) are not equivalent to \(yx\) on their own, the expressions manipulate equality for \(x\) and \(y\) under certain circumstances. Overall, \(3xy\) equals \(3\) times the product of \(x\) and \(y\), showing their synergy through algebraic manipulation.

Therefore, we cannot strictly prove that these expressions are equivalent in terms of their outcome numerically as \(yx\) and \(3x\) don't equate with \(3xy\). However, they share common elements in their formation and can illustrate fundamental properties of multiplication.