The solution to this equation is x = 2 and y = 4, or x = -2 and y = -4.
To solve this equation, we can use the method of substitution. We can solve for one variable in terms of the other in the second equation (xy = 8), and then substitute that value into the first equation (x^2 + y^2 = 16). This will give us a quadratic equation in one variable, which we can then solve using the quadratic formula.
No, there are only two solutions for this equation. This is because the equation x^2 + y^2 = 16 represents a circle with a radius of 4, and the equation xy = 8 represents a hyperbola. These two graphs intersect at only two points, giving us two solutions.
If the value of xy is changed to a different number, the solutions for x and y will also change. This is because the value of xy affects the slope of the hyperbola, which in turn affects where it intersects with the circle. Therefore, the solutions will depend on the specific value of xy.
This equation is relevant in real life when dealing with circles and hyperbolas, such as in geometry or physics. For example, it can be used to calculate the intersection points between a circular object and a hyperbolic object, or to find the dimensions of a circular or hyperbolic structure.