There she goes again!!
For your sanity you need to IGNORE that woman's attempt to solve this system.


   4x + 2y + 2z = 8 ------ Multiplying eq (ii) by 2 ------ eq (iv)
        5x + 5z = 15 ----- Adding eqs (iv) & (i) 
       5(x + z) = 5(3)_____x + z = 3 ------ eq (v)
       - 2x + z = - 3 -----Adding eqs (i) & (iii) ------ eq (vi)
             3x = 6 ------ Subtracting eq (vi) from eq (v)
             

          2 + z = 3 ------- Substituting 2 for x in eq (v)
             z = 3 - 2 = 1
   
   2(2) + y + 1 = 4 ------- Substituting 2 for x and 1 for z in eq (ii)
      4 + y + 1 = 4
          y + 5 = 4
             y = 4 - 5 = - 1
 
   You can do the CHECKS!!