I am very new at manipulating tensors and I have the following equation:
$$A_{\mu \nu\tau} b^\mu c^\nu = g_{\tau \rho} d^\rho$$
where $\tau$ is the independent index and $g_{\tau \rho}$ the metric tensor is. Furthermore I've also used the Einstein convention.
I want to solve this equation for the contravariant vector $d$ but I'm not sure how I should get rid of the metric multiplying this vector. I thought of two possible solutions to get rid of the metric:
1) Divide by the metric (less likely to be correct)
2) Contract the whole equation with the metric with upper indices (very likely to be correct).
Which of the two options I presented above is the correct way to get rid of the metric?
Multiply by $g^{\sigma\tau}$ and use that $$ g^{ab}g_{bc} = \delta^a_c, $$ so the right-hand side becomes $ d^{\sigma} $, which is what you want. I'll leave the left-hand side to you, since you don't say if you mind raising the index on $A_{\mu\nu\tau}$.