Solving Ax=b when A is ill-conditioned amplifies noise catastrophically. Tikhonov regularization adds a penalty term: x_λ = argmin{‖Ax−b‖² + λ‖Lx‖²}. The solution x_λ = (AᵀA + λLᵀL)⁻¹Aᵀb trades data fidelity for smoothness. The L-curve method finds the optimal λ at the corner where both residual and solution norm are small.