To solve this system of equations, we can use the method of substitution or elimination.

1) We will use the method of substitution to solve the system of equations:
4x - 3y = 12 (equation 1)
x + y = 45 (equation 2)
y - 2x = 1 (equation 3)
x - 6y = 4 (equation 4)

From equation 2, we can express x in terms of y:
x = 45 - y

Substitute x = 45 - y into equations 1, 3, and 4:

4(45 - y) - 3y = 12
180 - 4y - 3y = 12
-7y = -168
y = 24

x = 45 - 24 = 21

Therefore, the solution is x = 21, y = 24.

2) Now let's check the solution in the remaining equations:

x + y = 45
21 + 24 = 45, the equation is satisfied.

y - 2x = 1
24 - 2(21) = 1
24 - 42 = 1
-18 = 1, this equation is not satisfied.

x - 6y = 4
21 - 6(24) = 4
21 - 144 = 4
-123 = 4, this equation is also not satisfied.

Therefore, the solution x = 21, y = 24 does not satisfy all the equations, so there may be an error in the calculations.