I don't understand how to solve for solving systems of linear equations using the Gaussian elimination method.

x - 3y + 4z = 1
4x - 10y + 10z = 4
-3x - 9y-5z = -6

could someone explain how this is done

I know how to solve these, but I doubt if that is the 'Gaussian' method...

x - 3y + 4z = 1
4x - 10y + 10z = 4
-3x - 9y-5z = -6

First equation multiplied by three:
3x - 9y + 12z = 3

Added to the third equation:
-18y + 7z = -3 ~~~~ a.

First equation multiplied by 4:
4x - 12y + 16z = 4

Subtracted from second equation:
2y - 6z = 0 ~~~~~ b.

Now, with equations a. and b. you can solve for the rest.