Starting system:

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

Multiply both sides of the top equation by ½ :

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

Subtract multiples of top equation from others to eliminate "x" :

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x -   y + 2z = 5         y -  4z = -10         2y - 5z = -11

Subtract multiples of middle equation from others to eliminate "y" :

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

Divide both sides of the bottom equation by 3 :

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x       - 2z = -5       y   - 4z = -10               z = 3

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

x       = 1       y   = 2         z = 3

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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