Lagrange Multipliers (and finding extrema of a function with two restraints)

[Black Orpheus]

I need to find the extrema of f(x,y,z)=x+y+z subject to the restraints of x^2 - y^2 = 1 and 2x+z = 1. So the gradient of f equals (1,1,1) =
lambda1(2x,-2y,0) + lambda2(2,0,1). Solving for the lambdas I found that lambda1 = -1/(2x) = -1/(2y), or x=y. But this isn't possible if x^2 - y^2 = 1. Does this mean that there are no absolute max or min, or am I doing something wrong?

 

[HallsofIvy]

It may mean that the max or min is on the boundary of the set.