I need help solving this Lagrange Multiplier Problem.

Question

This is Solution provided, but I do know they came up with $\mu$ and the values for $x$, $y$, and $z$. Any help would be appreciated. Please solve using Lagrange multipliers with the two given constraints.

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Answer 1

Elaborating on the solution: we have $f(x,y,z) = x+2y$, $g(x,y,z) = x+y+z$, and $h(x,y,z) = y^2+z^2$. Calculating gradients, we get $\nabla f = (1,2,0)$, $\nabla g = (1,1,1)$, and $\nabla h = (0,2y,2z)$. We arrive at the system of equations: $$ \nabla f = \lambda \nabla g+\mu\nabla h$$ Substituting in gradients: $$ (1,2,0)=\lambda(1,1,1)+\mu(0,2y,2z)$$ Arriving at the system of equations mentioned: $$ 1=\lambda \\ 2 = \lambda + 2\mu y \\ 0 = \lambda +2\mu z\\ 1 = x+y+z\\ 4 = y^2+z^2$$ Plugging $\lambda = 1$ into the second equation yields $1=2\mu y$. Similarly, for the third equation, we get $-1=2\mu z$, from which we see that $-y=z$ and thus $y^2=(-y)^2=z^2$. Using that for the last equality, we have: $2y^2 = 4$ and so $y = \pm \sqrt{2} = -z$. Revisiting the fourth equality, we have that $x=1$ since $y=-z \iff y+z=0$. Finally we can arrive at the optimum: $(1,\sqrt{2},-\sqrt{2})$ and $(1,-\sqrt{2},\sqrt{2})$. Hopefully that clears up how they arrived at their values for $x,y,z$!