Answer
$(2, 3, -2)$
Work Step by Step
Label the original equations:
1. $3a + b + c = 7$
2. $a + 3b - c = 13$
3. $b = 2a - 1$
Substitute the expression for $b$ given by equation $3$ into equation $1$:
$3a + (2a - 1) + c = 7$
Combine like terms. This will be equation $4$:
$5a + c - 1 = 7$
Move constants to the right side of the equation by adding $1$ to both sides of the equation:
$5a + c = 8$
Substitute the expression for $b$ given by equation $3$ into equation $2$:
$a + 3(2a - 1) - c = 13$
Distribute and multiply to simplify. This will be equation $5$:
$a + 6a - 3 - c = 13$
Combine like terms on the left side of the equation:
$7a - 3 - c = 13$
Move constants to the right side of the equation by adding $3$ to both sides of the equation:
$7a - c = 16$
Set up a system of equations made up of equations $4$ and $5$:
$5a + c = 8$
$7a - c = 16$
No modification is needed here because the $c$ variable in the two equations are the same, differing only in sign.
Add the two equations together:
$12a = 24$
Divide each side of the equation by $12$ to solve for $a$:
$a = 2$
Substitute this value for $a$ into equation $4$ to find the value of $c$:
$5(2) + c = 8$
Multiply to simplify:
$10 + c = 8$
Subtract $10$ from each side of the equation to solve for $c$:
$c = -2$
Substitute the values for $a$ and $c$ into equation $2$ to find the value of $b$:
$2 + 3b - (-2) = 13$
Combine like terms on the left side of the equation:
$4 + 3b = 13$
Subtract $4$ from both sides of the equation to move constants to the right side of the equation:
$3b = 9$
Divide both sides of the equation by $3$:
$b = 3$
The solution is $(2, 3, -2)$.
To check the solution, plug in the three values into one of the original equations. Use equation $2$:
$2 + 3(3) - (-2) = 13$
$2 + 9 + 2 = 13$
$13 = 13$
The sides are equal to one another; therefore, the solution is correct.