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

Let's first solve the system of equations using the substitution method:

From the first equation: 4x + 5y = 1
Rearranging the equation, we get: y = (1 - 4x) / 5

Substitute y in the second equation:
5x + 7(1 - 4x) / 5 = 5
5x + 7 - 28x / 5 = 5
Multiplying everything by 5 to get rid of the fractions:
25x + 35 - 28x = 25
25x - 28x = 25 - 35
-3x = -10
x = (-10 / -3)
x = 10 / 3

Now, substitute the value of x back into the first equation to find y:
4(10 / 3) + 5y = 1
40/3 + 5y = 1
5y = 1 - 40/3
5y = 3/3 - 40/3
5y = -37/3
y = -37 / (3 * 5)
y = -37 / 15

Therefore, the solution to the system of equations is:
x = 10/3 and y = -37/15.