I am going to rewrite the equations because I prefer them in alphabetical order:
1) w + y + z = 6
2) 2w + x + z = 8
3) 3w + x + y = 10
4) x + y + z = 9
Combine equations 1) and 2) to remove the w-term:
w + y + z = 6 ---> multiply by 2 ---> 2w 2y + 2z = 12
2w + x + z = 8 ---> multiply by -1 ---> -2w - x - z = -8
Add down the columns ---> -x + 2y + z = 4 (equation 5)
Combine equations 1) and 3) to remove the w-term:
w + y + z = 6 ---> multiply by 3 ---> 3w + 3y + 3z = 18
3w + x + y = 10 ---> multiply by -1 ---> -3w - x - y = -10
Add down the columns ---> -x + 2y + 3z = 8
Combine these two new equations:
-x + 2y + z = 4 ---> -x + 2y + z = 4
-x + 2y + 3z = 8 ---> multiply by -1 ---> x - 2y - 3z = -8
Add down the columns: ---> -2z = -4 ---> z = 2
Combine equations 5) and 4) to remove the x-term:
-x + 2y + z = 4
x + y + z = 9
Add down the columns: 3y + 2z = 13
Since z = 2: 3y + 2(2) = 13 ---> 3y + 4 = 13 ---> 3y = 9 ---> y = 3
Using equation 4): x + y + z = 9
Since z = 2 and y = 3: x + 3 + 2 = 9 ---> x + 5 = 9 ---> x = 4
Using equation 1): w + y + z = 6 ---> w + 3 + 2 = 6 ---> w + 5 = 6 ---> w = 1