Starting system: 

2x  2y + 4z = 10 3x  2y + 2z = 5 4x  2y + 3z = 9 


Multiply both sides of the top equation by ½ : 
x  y + 2z = 5 3x  2y + 2z = 5 4x  2y + 3z = 9 


Subtract multiples of top equation from others to eliminate "x" : 
x  y + 2z = 5 y  4z = 10 2y  5z = 11 


Subtract multiples of middle equation from others to eliminate "y" : 
x  2z = 5 y  4z = 10 3z = 9 


Divide both sides of the bottom equation by 3 : 
x  2z = 5 y  4z = 10 z = 3 


Add multiples of bottom equation to others to get final solution : 


If any false statement (like 0 = 1) had appeared above , then the system would have
no solution ,
or the solution would be
: { } = empty set = Ø