6x-2y=-6 can be simplified by dividing through by 2: 3x-y=-3.
Take the value of x=6y-19 in the equation: 3(6y-19)-y=-3.
18y-57-y=-3, so 17y=57-3=54. From this y=54/17 (or 3 3/17). Since 3x=y-3, 3x=54/17-3=(54-51)/17=3/17 and x=1/17.
Let's see if these values fit the equations: 6x-2y=-6 becomes 6/17-108/17=-102/17=-6. Correct!
6y-19 becomes 324/17-19=(324-323)/17=1/17=x. Correct!