What kind of software can be used to solves systems of equations? For example, I have to solve the following equations:
$$\left\{\begin{align*}
&x^2 + y^2 + z^2 = 1\\
&Ax + By + Cz = 0
\end{align*}\right.$$
for $y$ and $z$, where $x$, $A$, $B$ and $C$ are known values. I need this for production. I guess I could solve this by hand, but there are many other of those and, if did all manually, my production would stop. Is there a software that solves this kind of situation?

Hot off the Mathematica presses:
> Solve[{x^2 + y^2 + z^2 - 1 == 0, A x + B y + C z == 0}, {y, z}] // Simplify

$$\begin{align}
\Bigg\{\Bigg\{y & \to -\frac{A B x+\sqrt{-C^2 \left(A^2 x^2+B^2 \left(-1+x^2\right)+C^2 \left(-1+x^2\right)\right)}}{B^2+C^2}, \\
z & \to \frac{-A C^2 x+B \sqrt{-C^2 \left(A^2 x^2+B^2 \left(-1+x^2\right)+C^2 \left(-1+x^2\right)\right)}}{C \left(B^2+C^2\right)}\Bigg\}, \\
\Bigg\{y & \to \frac{-A B x+\sqrt{-C^2 \left(A^2 x^2+B^2 \left(-1+x^2\right)+C^2 \left(-1+x^2\right)\right)}}{B^2+C^2}, \\
z & \to -\frac{A C^2 x+B \sqrt{-C^2 \left(A^2 x^2+B^2 \left(-1+x^2\right)+C^2 \left(-1+x^2\right)\right)}}{C \left(B^2+C^2\right)}\Bigg\}\Bigg\} \\
\end{align}$$
I have intentionally not simplified these equations, so as to give you a feel what Mathematica solutions are like.
