Answer
$(-3, 1, -1)$
Work Step by Step
Label the original equations first:
1. $a + b + c = -3$
2. $3b - c = 4$
3. $2a - b - 2c = -5$
Add equations $1$ and $3$ and modify them such that one variable is the same in both equations but differs only in sign. Since equation $2$ is missing an $a$ term, modify the first and third equations so that the $a$ term can be eliminated as well. Multiply equation $1$ by $-2$:
4. $-2(a + b + c) = -2(-3)$
Use distributive property:
4. $-2a - 2b - 2c = 6$
Combine equations $3$ and $4$:
3. $2a - b - 2c = -5$
4. $-2a - 2b - 2c = 6$
Add the equations:
6. $-3b - 4c = 1$
Combine equations $2$ and $6$ to eliminate a variable:
2. $3b - c = 4$
6. $-3b - 4c = 1$
Add the two equations together:
$-5c = 5$
Divide both sides of the equation by $-5$:
$c = -1$
Substitute this $c$ value into equation $2$ to find the value for $b$:
2. $3b - (-1) = 4$
Use distributive property:
2. $3b + 1 = 4$
Subtract $1$ from both sides of the equation:
$3b = 3$
Divide both sides by $3$:
$b = 1$
Substitute the values for $b$ and $c$ into one of the original equations to find $a$. Let's use equation $1$:
1. $a + 1 + (-1) = -3$
Combine like terms on the left side of the equation:
$a + 0 = -3$
Subtract $0$ from each side of the equation:
$a = -3$
The solution is $(-3, 1, -1)$.
To check the solution, plug in the three values into one of the original equations. Use equation $3$:
$2(-3) - 1 - 2(-1) = -5$
$-6 - 1 + 2 = -5$
$-5 = -5$
The sides are equal to one another; therefore, the solution is correct.
