b) Solve the linear system below using the \( L U \) decomposition method.
\[
\begin{aligned}
x_{1}+x_{2}+3 x_{4} & =8 \\
2 x_{1}+x_{2}-x_{3}+x_{4} & =7 \\
3 x_{1}-x_{2}-x_{3}+2 x_{4} & =14 \\
-x_{1}+2 x_{2}+3 x_{3}-x_{4} & =-7
\end{aligned}
\]

Answer

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Answer

Solve for x using backward substitution: \( U x = y\)

Steps

Step 1 ：Perform LU Decomposition of the coefficient matrix A: \( A = L U \)

Step 2 ：Solve for y using forward substitution: \( L y = b\)

Step 3 ：Solve for x using backward substitution: \( U x = y\)