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

First, let's rewrite the equations in standard form:

-5a + 3b - 4c = 50
4a - b - 3c = -2
-3a + 4b + 3c = 14

Now, let's eliminate one variable at a time.

Let's eliminate "a" from the first two equations by multiplying the first equation by 4 and the second equation by 5:

-20a + 12b - 16c = 200
20a - 5b - 15c = -10

Adding these two equations together, we get:

7b - 31c = 190

Next, let's eliminate "a" from the first and third equations. Multiply the first equation by 3 and the third equation by 5:

-15a + 9b - 12c = 150
-15a + 20b + 15c = 70

Adding these two equations together, we get:

29b + 3c = 220

Finally, let's eliminate "b" from the last two equations. Multiply the second equation by 4 and the third equation by 5:

16a - 4b - 12c = -8
-15a + 20b + 15c = 70

Adding these two equations together, we get:

a + 3c = 62

Now, we have the following new system of equations:

7b - 31c = 190
29b + 3c = 220
a + 3c = 62

Solving this system of equations will give us the values of a, b, and c.