To solve the system of equations:

1. Start by multiplying the first equation by 3 and the second equation by 2 to eliminate the y variable:
3(2x + 2y) = 3(1)
2(3x - 3y) = 2(4)

Simplify:
6x + 6y = 3
6x - 6y = 8

2. Add the two equations together to eliminate the y variable:
(6x + 6y) + (6x - 6y) = 3 + 8

Simplify:
12x = 11

3. Divide both sides of the equation by 12 to solve for x:
x = 11/12

4. Substitute the value of x back into one of the original equations to solve for y. Let's use the first equation:
2(11/12) + 2y = 1

Simplify:
11/6 + 2y = 1

Subtract 11/6 from both sides:
2y = 1 - 11/6
2y = 6/6 - 11/6
2y = -5/6

5. Divide both sides of the equation by 2 to solve for y:
y = -5/12

Therefore, the solution to the system of equations is x = 11/12 and y = -5/12.