Find the number of real solutions of the system of equations.
x^2−y^2=z
y^2−z^2 =x
z^2−x^2 =y
To find the number of real solutions of the given system of equations, we can analyze each equation individually.
Equation 1: x^2 - y^2 = z
Since this equation involves variables raised to the power of 2, it suggests that we should consider the quadratic nature of the equation. By rearranging the terms, we get:
x^2 - z = y^2
This implies that for any value of z, we obtain a quadratic equation in terms of x and y. If we fix the value of z and consider this equation, we can analyze the discriminant (b^2 - 4ac) to determine the type and number of solutions. However, since z is also a variable, we need to consider multiple cases.
Similarly, we can rearrange the other two equations:
Equation 2: y^2 - z^2 = x
Equation 3: z^2 - x^2 = y
Now, let's analyze the possible cases:
Case 1: x, y, and z are all real numbers
In this case, we can analyze each equation separately to determine the possible solutions.
Case 2: Either x, y, or z is a complex number
If any one of x, y, or z is a complex number, we need to consider the impact on the other equations. Since the equations involve quadratic forms, it is possible for the presence of complex numbers to introduce complex solutions.
To fully determine the number of real solutions, we would need to solve the equations simultaneously and consider all possible cases.