What is the solution to the system of equations

The answer to the system of equations is x = 3, y = -2 and z = 5. 

In order to find this you can use elimination to create two equations with only x and y. First we will add equation one with equation 2 multiplied by 2. 

-x + 2y + 2z = 3
6x + 2y - 2z = 4

5x + 4y = 7

Then  we can add equation 2 with equation 3. 

3x + y - z = 2
2x + y + z = 9

5x + 2y = 11

Now we can use these two equations together to solve for y. It will be easiest if we multiply the second one by -1. 

5x + 4y = 7
-5x - 2y = -11

2y = -4

And then we can solve for y. 

2y = -4
y = -2

With that answer we can go back to any equation with just y and x and solve for x. 

5x + 4y = 7
5x + 4(-2) = 7
5x - 8 = 7
5x = 15
x = 3

Now we can use x and y in any equation to find z. 

2x + y + z = 9
2(3) + (-2) + z = 9
6 - 2 + z = 9
4 + z = 9
z = 5