Solve the following system of equations for all three variables.
minus, x, plus, 2, y, plus, z, equals, 0
−x+2y+z=
0
minus, x, plus, 2, y, plus, 3, z, equals, 0
−x+2y+3z=
0
x, minus, 5, y, minus, 2, z, equals, 9
x−5y−2z=
9

Let's solve the system of equations for the variables \( x \), \( y \), and \( z \). The system is:

\[
\begin{align*}
- x + 2y + z &= 0 \quad \text{(Equation 1)} \\
- x + 2y + 3z &= 0 \quad \text{(Equation 2)} \\
x - 5y - 2z &= 9 \quad \text{(Equation 3)}
\end{align*}
\]

### Step 1: Simplify the System

Let's start by subtracting Equation 1 from Equation 2 to eliminate \( x \):

\[
\begin{align*}
(-x + 2y + 3z) - (-x + 2y + z) &= 0 - 0 \\
2z - z &= 0 \\
z &= 0
\end{align*}
\]

### Step 2: Substitute \( z = 0 \) into Equations 1 and 3

Substituting \( z = 0 \) into Equation 1:

\[
-x + 2y + 0 = 0 \implies -x + 2y = 0 \implies x = 2y \quad \text{(Equation 4)}
\]

Substituting \( z = 0 \) into Equation 3:

\[
x - 5y - 2(0) = 9 \implies x - 5y = 9
\]

### Step 3: Substitute Equation 4 into the Modified Equation 3

Substitute \( x = 2y \) into \( x - 5y = 9 \):

\[
\begin{align*}
2y - 5y &= 9 \\
-3y &= 9 \\
y &= -3
\end{align*}
\]

### Step 4: Find \( x \) using Equation 4

Substituting \( y = -3 \) into Equation 4:

\[
x = 2(-3) = -6
\]

### Step 5: Verify the Solution

Substituting \( x = -6 \), \( y = -3 \), and \( z = 0 \) back into the original equations:

1. **Equation 1**: 
   \[
   -(-6) + 2(-3) + 0 = 6 - 6 + 0 = 0 \quad \text{(True)}
   \]

2. **Equation 2**: 
   \[
   -(-6) + 2(-3) + 3(0) = 6 - 6 + 0 = 0 \quad \text{(True)}
   \]

3. **Equation 3**: 
   \[
   -6 - 5(-3) - 2(0) = -6 + 15 + 0 = 9 \quad \text{(True)}
   \]

All three equations are satisfied.

### Solution

The solution to the system of equations is:
\[
x = -6, \quad y = -3, \quad z = 0
\]

Do you want details or have any questions?

Here are some related questions for further exploration:
1. How do you solve a system of linear equations using matrices?
2. What is the determinant of a matrix, and how is it used to solve systems of equations?
3. Can you explain the method of substitution for solving systems of equations?
4. How does the method of elimination differ from substitution in solving equations?
5. What are the possible outcomes when solving systems of equations (e.g., one solution, no solution, infinitely many solutions)?
6. How are systems of linear equations used in real-world applications?
7. Can you solve a system of equations graphically, and how is this done?
8. What role do eigenvalues and eigenvectors play in linear algebra and systems of equations?

**Tip:** Always verify the solution of a system by substituting the values back into the original equations to ensure accuracy.

Learn how to solve a system of linear equations step-by-step. This problem involves three equations with three variables (x, y, z), solved using the substitution and elimination methods. Suitable for high school students (Grades 9-12).

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