\[\left\{ \begin{array}{l}x_{1}-8x_{2}=-80,\\4x_{1}+4x_{2}-x_{3}=8,\\3x_{2}-7x_{3}-6x_{4}=109,\\3x_{3}+x_{4}=-21 \end{array} \right.\]

Решаем систему уравнений методом прогонки.

Система может быть решена этим методом, так как матрица системы трёхдиагональная.

\[\left(\begin{matrix}- \beta_{1} & \gamma_{1} & 0 & 0\\\alpha_{2} & - \beta_{2} & \gamma_{2} & 0\\0 & \alpha_{3} & - \beta_{3} & \gamma_{3}\\0 & 0 & \alpha_{4} & - \beta_{4}\end{matrix}\right)\]

Выпишим коэффициенты

\[\alpha=\left( 0;4;3;3 \right)^T\]

\[\beta=\left( -1;-4;7;-1 \right)^T\]

\[\gamma=\left( -8;-1;-6 \right)^T\]

\[\delta=\left( -80;8;109;-21 \right)^T\]

Прямой ход

\[P_1 = \frac{\gamma _1}{\beta _1} = \frac{-8}{-4} = 8;Q_1 = - \frac{\delta _1}{\beta _1} = - \frac{-80}{-1} = -80;\]

\[P_{2} = \frac{\gamma _{2}}{{\beta _{2}} - {\alpha _{2}}{P_{1}}} = \frac{ -1}{ -4 - 4 \cdot 8} = \frac{1}{36};\]

\[Q_{2} = \frac{{\alpha _{2}}{Q_{1}} - {\delta _{2}}}{{\beta _2} - {\alpha _{2}}{P_{1}}}= \frac{4 \cdot \left( -80 \right) -8}{ -4 - 4 \cdot 8} = \frac{82}{9};\]

\[P_{3} = \frac{\gamma _{3}}{{\beta _{3}} - {\alpha _{3}}{P_{2}}} = \frac{ -6}{ 7 - 3 \cdot \frac{1}{36}} = - \frac{72}{83};\]

\[Q_{3} = \frac{{\alpha _{3}}{Q_{2}} - {\delta _{3}}}{{\beta _3} - {\alpha _{3}}{P_{2}}}= \frac{3 \cdot \frac{82}{9} -109}{ 7 - 3 \cdot \frac{1}{36}} = - \frac{980}{83};\]

\[{Q_{4}} = \frac{{{\alpha _{4}} \cdot {Q_{3} - {\delta _{4}}}}}{{{\beta _{4}} - {\alpha _{4}} \cdot {P_{3}}}} = \frac{{3} \cdot {\left( - \frac{980}{83} \right)} - { \left( -21 \right) }}{{-1} - {3} \cdot {\left( - \frac{72}{83} \right)}} = -9\]

Обратный ход

\[x_{4} = {Q_{4}} = -9;\]

\[{x_{3}} = Q_{3}+P_{3}x_{4} = - \frac{980}{83} + \left( - \frac{72}{83} \right)\cdot \left( -9 \right) = -4;\]

\[{x_{2}} = Q_{2}+P_{2}x_{3} = \frac{82}{9} + \frac{1}{36}\cdot \left( -4 \right) = 9;\]

\[{x_{1}} = Q_{1}+P_{1}x_{2} = -80 + 8\cdot 9 = -8;\]

$\left\{ \begin{array}{l}x_{1}-8x_{2}=-80,\\4x_{1}+4x_{2}-x_{3}=8,\\3x_{2}-7x_{3}-6x_{4}=109,\\3x_{3}+x_{4}=-21 \end{array} \right.$ \newline \textbf{Решаем систему уравнений методом прогонки}. \newline Система может быть решена этим методом, так как матрица системы трёхдиагональная. \newline $\left(\begin{matrix}- \beta_{1} & \gamma_{1} & 0 & 0\\\alpha_{2} & - \beta_{2} & \gamma_{2} & 0\\0 & \alpha_{3} & - \beta_{3} & \gamma_{3}\\0 & 0 & \alpha_{4} & - \beta_{4}\end{matrix}\right)$ \newline Выпишим коэффициенты \newline $\alpha=\left( 0;4;3;3 \right)^T$ \newline $\beta=\left( -1;-4;7;-1 \right)^T$ \newline $\gamma=\left( -8;-1;-6 \right)^T$ \newline $\delta=\left( -80;8;109;-21 \right)^T$ \newline Прямой ход \newline $P_1 = \frac{\gamma _1}{\beta _1} = \frac{-8}{-4} = 8;Q_1 = - \frac{\delta _1}{\beta _1} = - \frac{-80}{-1} = -80;$ \newline $P_{2} = \frac{\gamma _{2}}{{\beta _{2}} - {\alpha _{2}}{P_{1}}} = \frac{ -1}{ -4 - 4 \cdot 8} = \frac{1}{36};$ \newline $Q_{2} = \frac{{\alpha _{2}}{Q_{1}} - {\delta _{2}}}{{\beta _2} - {\alpha _{2}}{P_{1}}}= \frac{4 \cdot \left( -80 \right) -8}{ -4 - 4 \cdot 8} = \frac{82}{9};$ \newline $P_{3} = \frac{\gamma _{3}}{{\beta _{3}} - {\alpha _{3}}{P_{2}}} = \frac{ -6}{ 7 - 3 \cdot \frac{1}{36}} = - \frac{72}{83};$ \newline $Q_{3} = \frac{{\alpha _{3}}{Q_{2}} - {\delta _{3}}}{{\beta _3} - {\alpha _{3}}{P_{2}}}= \frac{3 \cdot \frac{82}{9} -109}{ 7 - 3 \cdot \frac{1}{36}} = - \frac{980}{83};$ \newline ${Q_{4}} = \frac{{{\alpha _{4}} \cdot {Q_{3} - {\delta _{4}}}}}{{{\beta _{4}} - {\alpha _{4}} \cdot {P_{3}}}} = \frac{{3} \cdot {\left( - \frac{980}{83} \right)} - { \left( -21 \right) }}{{-1} - {3} \cdot {\left( - \frac{72}{83} \right)}} = -9$ \newline Обратный ход \newline $x_{4} = {Q_{4}} = -9;$ \newline ${x_{3}} = Q_{3}+P_{3}x_{4} = - \frac{980}{83} + \left( - \frac{72}{83} \right)\cdot \left( -9 \right) = -4;$ \newline ${x_{2}} = Q_{2}+P_{2}x_{3} = \frac{82}{9} + \frac{1}{36}\cdot \left( -4 \right) = 9;$ \newline ${x_{1}} = Q_{1}+P_{1}x_{2} = -80 + 8\cdot 9 = -8;$