Question: Solve this system of equations. Write the solution as an ordered triple.
3xplus3yplus z
equals
negative 4
xminus3yplus2z
equals
negative 6
8xminus2yplus3z
equals
negative 30
1. **State the problem:** Solve the system of equations:
$$\begin{cases} 3x + 3y + z = -4 \\ x - 3y + 2z = -6 \\ 8x - 2y + 3z = -30 \end{cases}$$
2. **Use substitution or elimination.** We will use elimination here.
3. From the first equation, express $z$:
$$z = -4 - 3x - 3y$$
4. Substitute $z$ into the second and third equations:
Second equation:
$$x - 3y + 2(-4 - 3x - 3y) = -6$$
$$x - 3y - 8 - 6x - 6y = -6$$
$$\cancel{x} - 3y - 8 - 6x - 6y = -6$$
$$-5x - 9y - 8 = -6$$
Add 8 to both sides:
$$-5x - 9y = 2$$
Third equation:
$$8x - 2y + 3(-4 - 3x - 3y) = -30$$
$$8x - 2y - 12 - 9x - 9y = -30$$
$$-x - 11y - 12 = -30$$
Add 12 to both sides:
$$-x - 11y = -18$$
Multiply both sides by $-1$:
$$x + 11y = 18$$
5. Now solve the system:
$$\begin{cases} -5x - 9y = 2 \\ x + 11y = 18 \end{cases}$$
Multiply the second equation by 5:
$$5x + 55y = 90$$
Add to the first equation:
$$(-5x - 9y) + (5x + 55y) = 2 + 90$$
$$46y = 92$$
$$y = \frac{92}{46} = 2$$
6. Substitute $y=2$ into $x + 11y = 18$:
$$x + 11(2) = 18$$
$$x + 22 = 18$$
$$x = 18 - 22 = -4$$
7. Substitute $x=-4$ and $y=2$ into $z = -4 - 3x - 3y$:
$$z = -4 - 3(-4) - 3(2) = -4 + 12 - 6 = 2$$
**Final solution:**
$$(x, y, z) = (-4, 2, 2)$$