6x+7y=53

8y-9y= -21
write two equations that have your two intergers as a solution.Addition and subtraction methods

You have already written two equations. What do you mean by "Your" (my) two integers?

Shouldn't your second equation be 8x - 9y = -21? The way you have written it,-7 = -21, so y = 21

Why don't you just ask for the solution to your two equations? (I assume you meant the first y to be x in the second equation)
24x + 28y = 212
24x - 27y = -63
Subtract the second from the first
55y = 275
y = 5