Solve the system analytically. \[ \begin{aligned} 4 x-2 y+2 z & =34 \\ 5 x-4 y-4 z & =-10 \\ x-y+5 z & =31 \end{aligned} \]

Step 1 ：Given the system of equations: \[\begin{aligned} 4x - 2y + 2z & = 34 \\ 5x - 4y - 4z & = -10 \\ x - y + 5z & = 31 \end{aligned}\]

Step 2 ：First, multiply the third equation by 4 and subtract it from the first equation. Then, multiply the third equation by 5 and subtract it from the second equation. This will give us two new equations.

Step 3 ：Solve these two new equations to find the values of y and z.

Step 4 ：Once we have the values of y and z, substitute them into any of the original equations to find the value of x.

Step 5 ：The solution to the system of equations is \(\boxed{x = 10, y = 9, z = 6}\).