The first step is to multiply the first row by \( a_0/b_0 \) and subtract it from the second row. This is known as the forward substitution step. We obtain then $$ a_i = 0, $$ $$ b_i = b_i - \frac{a_{i-1}}{b_{i-1}}c_{i-1}, $$ and $$ f_i = f_i - \frac{a_{i-1}}{b_{i-1}}f_{i-1}. $$ At this point the simplified equation, with only an upper triangular matrix takes the form $$ \left( \begin{array}{ccccc} b_0 & c_0 & & & \\ & b_1 & c_1 & & \\ & & \ddots & & \\ & & & b_{m-2} & c_{m-2} \\ & & & & b_{m-1} \end{array} \right)\left( \begin{array}{c} x_0 \\ x_1 \\ \vdots \\ x_{m-2} \\ x_{m-1} \end{array} \right)=\left( \begin{array}{c} f_0 \\ f_1 \\ \vdots \\ f_{m-2} \\ f_{m-1} \\ \end{array} \right) $$