Question 101338
first take these two equations:
{{{2x + y + z = 3}}}
{{{-x + 2y + 2z = 1}}} (multiply both sides by {{{2}}} in this one)
then we will have
{{{2x + y + z = 3}}}
{{{-2x + 4y + 4z = 2}}}
now we will eliminate {{{x}}} by adding them and it will be:

{{{(y+4y) +(z+4z)=3+2}}}
{{{5y +5z =5}}} now divide both sides by {{{5}}},and we will have:

{{{y+z=1}}} 
then we have
{{{y=-z+1}}} which we can put in third equation 
{{{x - y - 3z = -6}}}
 and we will have
  
{{{x +z -1 - 3z = -6}}}
{{{x +z -1 -3z =-6}}}
{{{x -2z =-6 +1}}}
{{{x -2z =-5}}}

from this equation
{{{x=2z-5}}}

since we have  {{{x = 2z - 5}}} and {{{y = -z + 1}}}, we can use them to calculate {{{z}}}:

let use the second equation
 {{{-x + 2y + 2z = 1}}}   
 substitute {{{x}}} and {{{y }}}

{{{-(2z-5) + 2(-z + 1)+ 2z = 1}}}
{{{-2z + 5 -2z + 2 + 2z = 1}}} 
{{{-2z= -6}}}
{{{z=-6/-2}}}
{{{z=3 }}} 
now is easy to calculate {{{x}}} and {{{y}}}

{{{x= 2z – 5}}}
{{{x = 2*3 – 5}}}
{{{x = 1}}}

{{{y = -z + 1}}}
{{{y = -3 + 1}}}
{{{y = -2}}}