Algebra 1 elimination problem

I have been trying and trying these problems over again for awhile now:
Can someone please explain to me how to know what to multiply both equations by?

1. 2x/3 + 3y/4 = -10
y + 34 = 5x/6

2. 2x/5 + 3y/4 = -41/5
3x/5 = 3y+2 / 4 + 7/10

Thank you!

(2/3)*x + (3/4)*y = - 10
- (5/6)*x + y = - 34

8*x + 9*y = - 120, multiply by 5
-5*x + 6*y = - 204, multiply by 8

40*x + 45*y = - 600
-40*x + 48*y = - 1632
---------------------------------
93*y = - 2232

y = - 24

- (5/6)*x + y = - 34
- (5/6)*x = - 10
x = 12

1. 2x/3 + 3y/4 = -10
3y/4 = -10 -2x/3
3y/4=-2/3(x+15)
y=-8/9(15+x)