What is the solution set for the system of equations 2xy = 0 and 3xy = 9?

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To solve the system of equations given by 2xy = 0 and 3xy = 9, we need to analyze each equation individually.

Step 1: Analyze the first equation

The first equation, 2xy = 0, implies that either x = 0, y = 0, or both. This means all points along the x-axis (y = 0) and y-axis (x = 0) are part of the solution set for this equation.

Step 2: Analyze the second equation

The second equation, 3xy = 9, can be simplified by dividing both sides by 3:

xy = 3

This equation represents a hyperbola when plotted on a graph, with points that satisfy this equation being those whose product (xy) equals 3.

Step 3: Find the intersection of the solutions

Now, we need to find the intersection of the solutions from both equations. The first equation lets us determine that either x or y must be zero. We can explore this further:

  • If x = 0, substituting into xy = 3 gives us:

    • 0 imes y = 3

    Since this is never true, there are no solutions when x = 0.

  • If y = 0, substituting into xy = 3 gives us:

    • x imes 0 = 3

    Again, this is never true, leading to no solutions when y = 0.

Conclusion

Since neither equation can be satisfied when substituting zero for one variable, it turns out that there are no points (x, y) that satisfy both equations simultaneously. Therefore, the solution set for the given system of equations is:

Solution Set: ∅ (The empty set)

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