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

From the first equation, we can isolate y by solving for y:
-5x - 7y = -32
-7y = 5x - 32
y = (5x - 32)/(-7)

Now we can substitute this expression for y into the second equation:
-6x - 2((5x - 32)/(-7)) = -38
-6x + 10x/7 - 64/7 = -38
Multiplying through by 7 to get rid of the fractions:
-42x + 10x - 64 = -266
-32x - 64 = -266
-32x = -202
x = 202/32
x = 6.3125

Now that we have found the value of x, we can substitute it back into the equation we found for y:
y = (5(6.3125) - 32)/(-7)
y = (31.5625 - 32)/(-7)
y = (-0.4375)/(-7)
y = 0.0625

Therefore, the solution to the system of equations is x = 6.3125 and y = 0.0625.