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 system of equations, we can use the method of substitution. We'll start by solving the first equation for z:
z = x^2 - y^2.
Now we can substitute this value of z into the second equation:
y^2 - (x^2 - y^2)^2 = x.
Expanding the expression (x^2 - y^2)^2 gives us:
y^2 - (x^4 - 2x^2y^2 + y^4) = x.
Rearranging the terms, we get:
2x^2y^2 - x^4 + y^4 + y^2 - x = 0.
Similarly, we can substitute the value of z into the third equation:
z^2 - x^2 = y.
Again, rearranging the terms, we have:
x^2 - z^2 - y = 0.
So now we have a system of equations in terms of x and y:
2x^2y^2 - x^4 + y^4 + y^2 - x = 0,
x^2 - z^2 - y = 0.
To find the solutions, we can use numerical methods or software, such as graphing calculators or computer algebra systems. By graphing the equations or using appropriate software, we can determine the number of real solutions.
In this case, since the equations are non-linear and involve higher powers, the solution set may contain multiple real solutions, no real solutions, or an infinite number of real solutions.