If we have

$\displaystyle f(x,y,z) = 3x + 2y +5z$

Subject to the following constraints:

$\displaystyle g(x,y,z) = x + 2y+z \leq 430$

$\displaystyle h(x,y,z) = 3x+2z \leq 460$

$\displaystyle i(x,y,z) = x+4y \leq 420$

and we want to maximize (or minimize) it, how would we solve this? My lecture is originally about the Simplex method. But I was wondering, could we use Lagrange method here? I tried it but after finding partial derivatives of f(x,y,z) and the constraints I only come up with a solution to the scalar multiples of the constraints and I don't know what to do after that:

$\displaystyle \nabla f(x,y,z) =\lambda \cdot g(x,y,z) + \theta \cdot h(x,y,z) + \eta \cdot i(x,y,z) $

I get:

$\displaystyle \lambda = 5 , \theta = 0 , \eta = -2$