x = x^2 - y^2 -1
y = 2y
You asked:
Solve the system of equations:\[\begin{aligned}x &= {x}^{2} - {y}^{2} - 1\\y &= 2 y\end{aligned}\]
MathBot Answer:
The system of equations has \(2\) solutions.
\[x = \frac{1}{2} - \frac{\sqrt{5}}{2}, y = 0\]\[x = \frac{1}{2} + \frac{\sqrt{5}}{2}, y = 0\]
Solve \(y = 2 y\) for \(y\). \[\begin{aligned}y &= 2 y\\- y &= 0\\y &= 0\end{aligned}\]Substitute \(0\) for \(y\) in \(x = x^{2} - y^{2} - 1\) and solve.$$\begin{aligned}x &= x^{2} - y^{2} - 1\\x &= x^{2} - 0^{2} - 1\\- x^{2} + x + 1 &= 0 \\ x &= \frac{-(1) \pm \sqrt{(1)^{2} - 4(-1)(1)}}{2(1)} \\ x = \frac{1}{2} - \frac{\sqrt{5}}{2}&, x = \frac{1}{2} + \frac{\sqrt{5}}{2}\end{aligned}$$This yields the following solutions. $$\begin{aligned}x = \frac{1}{2} - \frac{\sqrt{5}}{2},\,y = 0\\ x = \frac{1}{2} + \frac{\sqrt{5}}{2},\,y = 0\end{aligned}$$