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

Let's start by using the substitution method.

From the first equation, we can solve for x:
5x - 5y - 4z = -3
5x = 5y + 4z - 3
x = (5y + 4z - 3)/5

Now we substitute x into the other two equations:

x - y + 5z = 11
(5y + 4z - 3)/5 - y + 5z = 11
5y + 4z - 3 - 5y + 5z = 55
4z - 3 + 5z = 55
9z - 3 = 55
9z = 58
z = 58/9

Next, we substitute z back into the equation for x:
x = (5y + 4z - 3)/5
x = (5y + 4(58/9) - 3)/5
x = (5y + 232/9 - 3)/5
x = (5y - 9 + 232)/45
x = (5y + 223)/45

Now, we can substitute both x and z into the third equation:

4x - 3y - 6z = -9
4(5y + 223)/45 - 3y - 6(58)/9 = -9
(20y + 892)/45 - 3y - 348/3 = -9
20y + 892 - 135y - 1044 = -405
-115y - 152 = -405
-115y = -253
y = 253/115

Now that we have found the values of y and z, we can substitute them back into the equations for x:

x = (5(253/115) + 223)/45
x = (1265/115 + 223)/45
x = (1265 + 2585)/1035
x = 3850/1035

Therefore, the solution to the system of equations is:
x = 3850/1035
y = 253/115
z = 58/9