Final answer:
Yes, x * y and x · y are equivalent expressions for multiplication of any two real numbers, x and y. Multiplication rules for signs apply, and the operation maintains equality when applied to both sides of an equation.
Step-by-step explanation:
The question asks whether the expression x * y is equivalent to the expression x · y for all real numbers x and y. The answer is yes, these expressions are equivalent because both represent the multiplication of x and y. Multiplication is a binary operation that combines two numbers to make a new number, known as the product.
It's also important to understand the properties of multiplication, especially concerning the signs of the numbers involved. When two positive numbers multiply, such as 2x3, the product is positive (6). When two negative numbers multiply, such as (-4) x (-3), the product is also positive (12). However, when numbers of opposite signs are multiplied, such as (-3) x 2 or 4 x (-4), the product is negative (-6 and -16 respectively).
When multiplying or dividing both sides of an equation by the same number, equality is maintained, and the sign rules for multiplication apply. This technique is often used to simplify equations and solve for unknown variables.