Answer
$(6, 0, -2)$
Work Step by Step
Label the original equations first:
1. $3x + y - 2z = 22$
2. $x + 5y +z = 4$
3. $x = -3z$
The first step is to substitute the expression for $x$ into both the first and second equations to eliminate one variable. Label these the fourth and fifth equations:
4. $3(-3z) + y - 2z = 22$
5. $-3z + 5y +z = 4$
Multiply to simplify.
4. $-9z + y - 2z = 22$
5. $-3z + 5y +z = 4$
Combine like terms:
4. $-11z + y = 22$
5. $-2z + 5y = 4$
Modify these two equations such that one variable is the same in both equations but differs only in sign.
Multiply the fourth equation by $-5$:
6. $55z - 5y = -110$
Combine this equation and the fifth equation to eliminate the $x$ variable:
5. $-2z + 5y = 4$
6. $55z - 5y = -110$
Add the equations together:
$53z = -106$
Divide both sides of the equation by $53$:
$z = -2$
Substitute this $z$ value into the third equation to find the value of $x$:
3. $x = -3(-2)$
Multiply to solve for $x$:
$x = 6$
Substitute the $x$ and $z$ values into the second equation to find $y$:
2. $6 + 5y +(-2) = 4$
Combine like terms on the left side of the equation:
$$5y + 4 = 4$$
Subtract $4$ from each side of the equation:
$5y = 0$
Divide each side of the equation by $5$:
$$y = 0$$
Our solution is $(6, 0, -2)$.