−x	 + 	2y	 − 	z	 = 	0
−x	 − 	y	 + 	2z	 = 	0
2x			 − 	z	 = 	3


You have a system of linear equations:

\[
\begin{aligned}
1. & \ -x + 2y - z = 0 \\
2. & \ -x - y + 2z = 0 \\
3. & \ 2x - z = 3
\end{aligned}
\]

### Solution Steps:

#### Step 1: Solve equation 3 for \( z \)
From equation (3):
\[
2x - z = 3 \implies z = 2x - 3
\]

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

- **Substitute into equation (1):**

\[
-x + 2y - (2x - 3) = 0
\]
Simplifying:
\[
-x + 2y - 2x + 3 = 0 \implies -3x + 2y = -3 \implies 3x - 2y = 3 \quad \text{(Equation 4)}
\]

- **Substitute into equation (2):**

\[
-x - y + 2(2x - 3) = 0
\]
Simplifying:
\[
-x - y + 4x - 6 = 0 \implies 3x - y = 6 \quad \text{(Equation 5)}
\]

#### Step 3: Solve equations (4) and (5)
You now have:
\[
\begin{aligned}
4. & \ 3x - 2y = 3 \\
5. & \ 3x - y = 6
\end{aligned}
\]

Solve equation (5) for \( y \):
\[
y = 3x - 6
\]

#### Step 4: Substitute \( y = 3x - 6 \) into equation (4)
Substitute into equation (4):
\[
3x - 2(3x - 6) = 3
\]
Simplifying:
\[
3x - 6x + 12 = 3 \implies -3x = -9 \implies x = 3
\]

#### Step 5: Find \( y \) and \( z \)
- Substitute \( x = 3 \) into \( y = 3x - 6 \):
\[
y = 3(3) - 6 = 9 - 6 = 3
\]

- Substitute \( x = 3 \) into \( z = 2x - 3 \):
\[
z = 2(3) - 3 = 6 - 3 = 3
\]

### Final Solution:
\[
x = 3, \quad y = 3, \quad z = 3
\]

Would you like more details on any step?

Here are 5 related questions you might find useful:
1. How do you solve systems of linear equations using matrices?
2. What is substitution method in solving systems of equations?
3. How does elimination method differ from substitution in solving equations?
4. Can a system of linear equations have no solution or infinite solutions?
5. What are the real-life applications of solving systems of equations?

**Tip**: When solving systems of equations, always simplify expressions at every step to avoid errors!

−x + 2y − z = 0
−x − y + 2z = 0
2x − z = 3

Learn how to solve a system of linear equations using the substitution method. The problem involves three variables and key concepts in algebra, including substitution and simplifying linear equations. Suitable for students in Grades 9-11.

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