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If I want to solve $\mathbf A \mathbf x = \mathbf b$, but I am only interested in the value of $x_1$, what algorithm should I use, and will it always be strictly more efficient than solving for all of $\mathbf x$?

This can be useful in finding the hitting time in a Markov chain. If we want to calculate the expected hitting time from $0$ to $j$, we can solve a system involving all the expected hitting times from $i$ to $j$ as unknowns, but we only need $i = 0$. However, in some cases the matrix involved is triangular or almost triangular; intuition tells me in those cases, it'd be the same to solve for all of them.

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Follow the Gaussian elimination and manipulate the matrix $A$ such that in one row only the coefficient of $x_1$ survives!

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