linguistic:paraphrasing
Forward substitution

The matrix equation L x = b can be written as a system of linear equations

Observe that the first equation (ℓ 1, 1 x 1 = b 1 {\displaystyle \ell _{1,1}x_{1}=b_{1}}) only involves x 1 {\displaystyle x_{1}}, and thus one can solve for x 1 {\displaystyle x_{1}} directly. The second equation only involves x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}}, and thus can be solved once one substitutes in the already solved value for x 1 {\displaystyle x_{1}}. Continuing in this way, the k {\displaystyle k} -th equation only involves x 1, …, x k {\displaystyle x_{1},\dots,x_{k}}, and one can solve for x k {\displaystyle x_{k}} using the previously solved values for x 1, …, x k − 1 {\displaystyle x_{1},\dots,x_{k-1}}. The resulting formulas are:

A matrix equation with an upper triangular matrix U can be solved in an analogous way, only working backwards.