To solve the system of equations:

1) X - 4y = 3
2) x + y = 4

We can use the substitution method or the elimination method.

Substitution method:
From equation 2, we can express x in terms of y:
x = 4 - y

Then, substitute this value of x into equation 1:
4 - y - 4y = 3
4 - 5y = 3
-5y = -1
y = 1

Now that we have found the value of y, we can substitute it back into equation 2 to find x:
x + 1 = 4
x = 3

Therefore, the solution to the system of equations is x = 3 and y = 1.