solving a system in five unknowns for lagrange multiplier

1. The problem statement, all variables and given/known data



I have to find the extrema of a given function with two constraints



[tex]f(x,y,z) = x+y+z;x^2-y^2=1;2x+z=1[/tex]











3. The attempt at a solution

If I create a new function F



then I have



[tex]F(x,y,z,\lambda,\mu)=x+y+z-(x^2\lambda - y^2\lambda -\lambda) -(2x\mu + z\mu - \mu) [/tex]



and taking the partials



[tex]\left\{\begin{array}{cc} F_{x} = 1-2x\lambda - 2\mu =0\\ F_{y} = 1+2y\lambda = 0 \\F_{z} = 1 - \mu = 0 \\ F_{\lambda} = -x^2 + y^2 + 1 = 0 \\ F_{\mu} = -2x - z +1 = 0 \end{array}\right. [/tex]



so now,



[tex]\mu =1[/tex]

solving for lambda



[tex]\lambda = -\frac{1}{2y} = \frac{1-2}{2x}[/tex]



now solving for x [or y] [tex]x=y[/tex]



but this causes an issue with [tex]-x^2+y^2+1=0[/tex] because 1 does not equal zero