Finding the value of an expression given three equations. $$x+y+2z=22$$
$$3x-2y+z=6$$
$$7x+3y-5z=1$$
above are three equations and we have to find the value of $$x^2+y^2+z^2=?$$

Hint: $$\det \begin{vmatrix}1&1&2\\3&-2&1\\7&3&-5\end{vmatrix}\neq 0$$ so the system has a unique solution. Find it and substitute in $x^2+y^2+z^2$. To check you should find $(x,y,z)=(3,5,7)$. Can you solve the system of three equations with the three unknowns (by performing elementary row operations)?
