The partial derivatives of L with respect to E, X, Z, and give four equations with

The partial derivatives of L with respect to E, X, Z, and λ give four equations with four unknowns. Although these first-order conditions cannot be solved analytically, they can be rearranged so that E, X, and Z are each expressed as a function of λ and parameters. A solution can be found by guessing values of λ until a value is found that implies values of E, X, and Z that satisfy the budget constraint.