solve the system: 3x-3y+z=10        3x+2y-3z=-2        -3x+z=-2

We are given the following system of equations:

\[
\begin{aligned}
1. \quad & 3x - 3y + z = 10 \\
2. \quad & 3x + 2y - 3z = -2 \\
3. \quad & -3x + z = -2 \\
\end{aligned}
\]

We can solve this system step by step using substitution or elimination. Let's proceed with substitution by solving the third equation for \( z \) and substituting it into the first and second equations.

### Step 1: Solve Equation 3 for \( z \)
\[
-3x + z = -2
\]
\[
z = 3x - 2
\]

### Step 2: Substitute \( z = 3x - 2 \) into Equations 1 and 2.

#### Substitute into Equation 1:
\[
3x - 3y + (3x - 2) = 10
\]
Simplify:
\[
6x - 3y - 2 = 10
\]
\[
6x - 3y = 12
\]
\[
2x - y = 4 \quad \text{(Equation 4)}
\]

#### Substitute into Equation 2:
\[
3x + 2y - 3(3x - 2) = -2
\]
Simplify:
\[
3x + 2y - 9x + 6 = -2
\]
\[
-6x + 2y = -8
\]
\[
-3x + y = -4 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of two equations (Equations 4 and 5).

\[
\begin{aligned}
1. \quad & 2x - y = 4 \\
2. \quad & -3x + y = -4 \\
\end{aligned}
\]

Add these two equations to eliminate \( y \):

\[
(2x - y) + (-3x + y) = 4 + (-4)
\]
\[
-1x = 0
\]
\[
x = 0
\]

### Step 4: Substitute \( x = 0 \) into Equation 4 to find \( y \).

\[
2(0) - y = 4
\]
\[
-y = 4
\]
\[
y = -4
\]

### Step 5: Substitute \( x = 0 \) into the expression for \( z \) (from Step 1).

\[
z = 3(0) - 2 = -2
\]

### Final Solution:
The solution to the system of equations is:
\[
x = 0, \quad y = -4, \quad z = -2
\]

Would you like further details or clarifications on any of the steps?

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Here are 5 related questions to explore:

1. How do you solve systems of linear equations using the substitution method?
2. What are the advantages of using elimination versus substitution?
3. How can matrix methods, such as Gaussian elimination, solve systems of equations?
4. What are the conditions for a system of equations to have no solution or infinitely many solutions?
5. How would the solution change if one of the constants in the equations was different?

**Tip:** Always check your solutions by substituting them back into the original system of equations to ensure they satisfy all the equations.

Solve the system: 3x - 3y + z = 10, 3x + 2y - 3z = -2, -3x + z = -2

Learn how to solve the system of equations 3x - 3y + z = 10, 3x + 2y - 3z = -2, and -3x + z = -2 using substitution and elimination methods. This detailed step-by-step guide is ideal for high school students (Grades 9-12) and covers essential linear algebra concepts.

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