Find the Domain (xy)/2=y

Problem

(x*y)/2=y

Solution

1. Identify the nature of the expression. The task asks for the domain of the equation (x*y)/2=y In the context of an equation relating x and y the domain refers to the set of all possible values of x for which a corresponding y exists.

2. Rearrange the equation to group all terms on one side to analyze the relationship between the variables.

(x*y)/2−y=0

3. Factor out the common term y from the expression.

y*(x/2−1)=0

4. Apply the zero product property. This equation is satisfied if either y=0 or x/2−1=0

y=0

x/2=1⇒x=2

5. Analyze the constraints on x The equation represents the union of two lines: the horizontal line y=0 (the x-axis) and the vertical line x=2 For any real number x there is at least one corresponding value of y Specifically, if x≠2 then y must be 0 If x=2 then y can be any real number.

6. Determine the domain. Since there are no values of x that result in undefined operations (like division by zero or square roots of negative numbers) and every x has a corresponding y the domain is all real numbers.

Final Answer

Domain:(−∞,∞)

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