I'm trying to find the max and mins of the equation $f(x,y,z) = xy + 3xz + 2yz$ on the constraint, $g(x,y,z)=5x+9y+z-10$. So according to the Lagrange Multiplier procedure, I take the partial derivatives of both equations and get,
$\frac{\partial f}{\partial x} = y + 3z$, $\frac{\partial g}{\partial x} = 5$
$\frac{\partial f}{\partial y} = x + 2z$, $\frac{\partial g}{\partial y} = 9$
$\frac{\partial f}{\partial z} = 3x + 2y$, $\frac{\partial g}{\partial z} = 1$
Then, using $\nabla f = \lambda\nabla g$
$y+3z = 5\lambda$
$x+2z = 9\lambda$
$3x+2y = \lambda$
However, the problem I encounter now is that I am having trouble getting the variables x, y, and z in terms of one variable when solving for lambda. If I am unable to do so, does this mean that the maxima and minima are undefined? I also noticed that the constraint was a plane, so perhaps there can't be any extrema because the constraint equation runs on values of x, y, and z that can take range from $\infty$ to $-\infty$ (no restriction on domain)?