\begin{align*} x\lambda_{1}&=-\frac{1}{2}\\ y\lambda_{2}&=\frac{1}{2}\\ z\lambda_{1}+z\lambda_{2}&=1\\ x^{2}+z^{2}&=1\\ y^{2}+z^{2}&=1\\ \end{align*}
I wonder if there's any simple method of solving such systems of equations...is there?
I've been always doing this just intuitively, but there are too much equations here, I get stuck at $x^{2}=y^{2}$...
For a system of quadratic equations, as pointed out by Tunococ there is no general method you'll want to carry out by hand. Mathematica and other software can be used to easily find the solutions, and if any old numerical solution will do (you don't need to find all of the solutions, and a very good approximation is OK i.e. you don't need the solution in closed form) a simple implementation of Newton's method often works.
Otherwise, you'll want to take advantage of special structure of the problem. For instance, you can see immediately from your last two equations that $x = \pm y$.
If $x=0$ the first equation is impossible. So $x\neq 0$. Similarly, let's take the plus case first. Then $\lambda_1 = -\lambda_2$ and the third equation is impossible. So we must have $x = -y$ and $\lambda_1 = \lambda_2$.
Eliminating lambda gives us
\begin{align*} \frac{2z}{-2x} &= 1\\ x^2+z^2 &= 1, \end{align*} in other words $x=z = \pm\frac{1}{\sqrt{2}}.$