I am would like to find:
$$\max U=4x_1+3x_2 \\ \text{s.t. } 2x_1+x_2 \leq 10\\ \text{and } x_1,x_2 \geq 0$$
Using Lagrange:
$$L =4x_1+3x_2 -\lambda_1(2x_1+x_2-10) + \lambda_2(x_1) + \lambda_3(x_2)$$
My idea is to use first-order conditions, for example:
$$\frac{\partial L}{\partial x_1} = 4-2\lambda_1+\lambda_2 = 0$$
. . .etc
Is this method feasible? Or does it make more sense to use linear programming to solve?