Challenge: Solving a Linear System with LU Decomposition

A student is analyzing a simple network where the flow balance can be represented as a system of linear equations:

Where:

• $A$ is a $3 \times 3$ coefficient matrix;
• $\vec{b}$ is a vector of known quantities;
• $\vec{x}$ is the vector of unknowns to be determined.

Your goal is to solve for $\vec{x}$ by performing an LU decomposition of matrix $A$, followed by forward and backward substitution. Finally, you'll compare your computed result with NumPy’s built-in solver to confirm correctness.

Your task:

1. Complete the Python code to:
   • Perform LU decomposition by filling in the missing expressions for $L$ and $U$.
   • Implement forward substitution to solve $L\vec{y} = \vec{b}$.
   • Implement backward substitution to solve $U\vec{x} = \vec{y}$.
2. Compare your result with np.linalg.solve() to verify accuracy.