Answer
$(-2, -1, 12)$
Work Step by Step
Label the original equations:
1. $4a + 2b + c = 2$
2. $5a - 3b + 2c = 17$
3. $a - 5b = 3$
Solve equation $3$ for $a$. This will be equation $4$:
4. $a = 5b + 3$
Substitute the expression for $a$ given by equation $4$ into equation $1$:
$4(5b + 3) + 2b + c = 2$
Distribute and multiply to simplify. This will be equation $5$:
5. $20b + 12 + 2b + c = 2$
Combine like terms:
5. $22b + 12 + c = 2$
Move constants to the right side of the equation by subtracting $12$ from both sides of the equation:
5. $22b + c = -10$
Substitute the expression for $a$ given by equation $4$ into equation $2$:
$5(5b + 3) - 3b + 2c = 17$
Distribute and multiply to simplify. This will be equation $6$:
6. $25b + 15 - 3b + 2c = 17$
Combine like terms on the left side of the equation:
6. $22b + 2c + 15 = 17$
Move constants to the right side of the equation by subtracting $15$ from both sides of the equation:
6. $22b + 2c = 2$
Set up a system of equations made up of equations $5$ and $6$:
5. $22b + c = -10$
6. $22b + 2c = 2$
Modify equation $5$ so that the $b$ variable is the same in both equations but differs only in sign.
Multiply equation $5$ by $-1$:
5. $-1(22b + c) = -1(-10)$
6. $22b + 2c = 2$
Distribute and multiply to simplify:
$-22b - c = 10$
$22b + 2c = 2$
Add the two equations together:
$c = 12$
Substitute this value for $c$ into equation $5$ to find the value of $b$:
$22b + 12 = -10$
Subtract $12$ from each side of the equation to move constants to the right side of the equation:
$22b = -22$
Divide each side of the equation by $22$ to solve for $b$:
$b = -1$
Substitute the values for $b$ and $c$ into equation $1$ to find the value of $a$:
$4a + 2(-1) + 12 = 2$
$4a + 10 = 2$
$4a = -8$
Divide both sides of the equation by $4$ to solve for $a$:
$a = -2$
The solution is $(-2, -1, 12)$.
