# Analytical solution, equation of 5 variables with quadratic components

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.

• OK, now divide the first three equations by $\lambda_1$, and call $1\over\lambda_1$ and $\lambda_2\over\lambda_1$ your new variables. You'll still have five equations, and all but one will be linear. Use the linear equations to exclude all but one variable, and solve the remaining quadratic equation. – Ivan Neretin Aug 31 '16 at 18:12

You have the equations

$$1 = 2\lambda_1x + \lambda_2\tag{1}$$

$$1 = 2\lambda_1y - \lambda_2\tag{2}$$

$$1 = 2\lambda_1z - \lambda_2\tag{3}$$

$$x^2 + y^2 + z^2=1\tag{4}$$

$$x-y-z=1\tag{5}$$

Adding $(1)$ and $(2)$ gives

$$1=\lambda_1(x+y)\tag{6}$$

Adding $(1)$ and $(3)$ gives

$$1=\lambda_1(x+z)$$

From these two equations we get $x+y=x+z$ so that $y=z$.

Substituting into $(5)$ and rearranging gives

$$x=1+2y\tag{7}$$

Substituting this into $(4)$ gives

$$(1+2y)^2+y^2+y^2=1$$

which has solutions $y=0$ and $y=-2/3$.

• If $y=z=0$ then $(7)$ gives $x=1$. Equations $(6)$ and $(2)$ give $\lambda_1=1$, and $\lambda_2=-1$.
• If $y=z=-2/3$ then $(7)$ gives $x=-1/3$. Equations $(6)$ and $(2)$ give $\lambda_1=-1$ and $\lambda_2=1/3$.