8x-3y=22 multiply through by 6 → 48x - 18y = 132 (1)
6x+2y=12 multiply through by -8 → -48x -16y = -96 (2)
Add (1) and (2) and we get
-34y = 36 divide through by -34
y = -36/34 = -18/17
And using any of the equations to find x we have
6x + 2(-18/17) = 12
6x - 36/17 = 12
6x = 12 + 36/17 divide both sides by 6
x = 2 + 6/17 = 40/17