Given the function $f = x + y + z$ and conditions $g = 0 = x^2 + y^2 + z^2 -1$ and $h =x-y-z-1=0$ one can find the max/min of the function $f$ with Lagrange's method, leading to the set of equations:
$1 = 2\lambda_1x + \lambda_2$
$1 = 2\lambda_1y - \lambda_2$
$1 = 2\lambda_1z - \lambda_2$
$0 = x^2 + y^2 + z^2 - 1$
$0 = x-y-z -1$
I can solve this numerically in matlab but how (if at all possible) would you solve this analytically? There is no way I know of to set this up as $Ax = b$, there is a quadratic equation in all 3 variables and $\lambda$ is coupled with $x,y,z$.
Edit: Ty for pointing out in comments the incorrect sign.