Final answer:
To solve the system of equations, we can use the method of substitution. By substituting the values of a and b in terms of c, and solving for c, we find c = -1. Substituting this value of c back into the equations, we can find the values of a and b which are a = -10 and b = -7/4 respectively.
Step-by-step explanation:
To solve the system of equations:
-a - 3 + 4c = 3
5a - 8b + 5c = 27
5a - 2b + 6c = 1
We can solve this system using the method of elimination or substitution. Let's use the method of substitution.
From the first equation, we can solve for a: a = -6 + 4c
Substitute this value of a in the second equation: 5(-6 + 4c) - 8b + 5c = 27
Simplify and solve for b: -30 + 20c - 8b + 5c = 27
Combine like terms: -30 + 27 + 20c + 5c - 8b = 0
Simplify further: -3 + 25c - 8b = 0
From the third equation, solve for a: a = -5 + 2b - 6c
Substitute this value of a in the second equation: 5(-5 + 2b - 6c) - 8b + 5c = 27
Simplify and solve for c: -25 + 10b - 30c - 8b + 5c = 27
Combine like terms: -25 + 27 + 10b - 8b - 30c + 5c = 0
Simplify further: 2b - 25c + 2 = 0
We now have the system of equations:
-3 + 25c - 8b = 0
2b - 25c + 2 = 0
We can solve this system of equations using the method of elimination.
Multiply the first equation by 2: -6 + 50c - 16b = 0
Add this equation to the second equation: -6 + 50c - 16b + 2b - 25c + 2 = 0
Combine like terms: -4b + 25c - 4 = 0
Solve for b: b = (25c - 4)/4
Substitute this value of b in the first equation: -3 + 25c - 8((25c - 4)/4) = 0
Simplify and solve for c: -3 + 25c - 50c + 8 = 0
Combine like terms: 5c - 3 + 8 = 0
Simplify further: 5c + 5 = 0
Solve for c: c = -1
Substitute this value of c in the first equation: -a - 3 + 4(-1) = 3
Simplify and solve for a: -a - 3 - 4 = 3
Combine like terms: -a - 7 = 3
Solve for a: a = -10
The solution to the system of equations is a = -10, b = (25(-1) - 4)/4 = -7/4, and c = -1. Therefore, the correct option is a = -10, b = -7/4, c = -1.