Here is the energy and angular momentum for the Schwarzschild metric, a spherically symmetric spacetime in General Relativity.
$E=-u_{t}=-g_{ta}u^{a}=-g_{tt}u^t-g_{t\phi}u^{\phi}$
$L=u_{\phi}=g_{\phi a}u^{a}=g_{\phi t}u^t+g_{\phi\phi}u^{\phi}$
I want to solve this system for $u^t$ and $u^{\phi}$.But I dont know how to "leave" $u^t$ and $u^{\phi}$ alone. All $g_{\alpha\beta}$ are the metric components of this spacetime.
I tried to multiply the first equation with $g^{ta}$ to eliminate $g_{tt}$ from $g_{tt}u^{t}$ :
$g^{ta}E=-g^{ta}g_{tt}u^{t}-g^{ta}g_{t\phi}u^{\phi}=-\delta^{a}_{t}u^t-\delta^{a}_{\phi}u^{\phi}=-2u^{a}=g^{at}E$
and then i change $a$ with $t$ because $a$ is a dummy index so i get : $-2u^t=g^{tt}E$ --->$u^t=1/2*g^{tt}E$. But that it is not the solution.
Edit: E and L expressions come from an axisymmetric spacetime.That is why there are mixed metric indices.
This looks like an ordinary linear system of equations: $$\begin{cases} -g_{tt}u^t-g_{t\phi}u^{\phi} = E \\ g_{\phi t}u^t+g_{\phi\phi}u^{\phi} = L \end{cases}$$ or in matrix form: $$\begin{pmatrix} -g_{tt} & -g_{t\phi} \\ g_{\phi t} & g_{\phi\phi} \end{pmatrix} \begin{pmatrix} u^t \\ u^\phi \end{pmatrix} = \begin{pmatrix} E \\ L \end{pmatrix} $$
The solution is $$ \begin{pmatrix} u^t \\ u^\phi \end{pmatrix} = \frac{1}{-g_{tt}g_{\phi\phi}+g_{\phi t}g_{t\phi}} \begin{pmatrix} g_{\phi\phi} & g_{t\phi} \\ -g_{\phi t} & -g_{tt} \end{pmatrix} \begin{pmatrix} E \\ L \end{pmatrix} $$ i.e. $$\begin{cases} u^t = (g_{\phi\phi} E + g_{t\phi} L)/(g_{\phi t} g_{t \phi} - g_{tt} g_{\phi\phi}) \\ u^\phi = (-g_{\phi t} E - g_{tt} L)/(g_{\phi t} g_{t \phi} - g_{tt} g_{\phi\phi}) \end{cases}$$