Solve using elimination, x+y-2z=5, -x+2y+z=2, 2x+3y-z=9

We will solve the system of equations using the **elimination method**. The system is:

\[
\text{(1)} \quad x + y - 2z = 5
\]
\[
\text{(2)} \quad -x + 2y + z = 2
\]
\[
\text{(3)} \quad 2x + 3y - z = 9
\]

### Step 1: Add equations (1) and (2) to eliminate \(x\)
\[
(x + y - 2z) + (-x + 2y + z) = 5 + 2
\]
Simplifying:
\[
y + 2y - 2z + z = 7
\]
\[
3y - z = 7 \quad \text{(Equation A)}
\]

### Step 2: Add equations (2) and (3) to eliminate \(x\)
\[
(-x + 2y + z) + (2x + 3y - z) = 2 + 9
\]
Simplifying:
\[
-x + 2x + 2y + 3y + z - z = 11
\]
\[
x + 5y = 11 \quad \text{(Equation B)}
\]

### Step 3: Solve for \(x\) and \(y\) using Equation B
From Equation B:
\[
x + 5y = 11 \quad \text{so} \quad x = 11 - 5y \quad \text{(Equation C)}
\]

### Step 4: Substitute \(x = 11 - 5y\) into one of the original equations
Substitute \(x\) into equation (1):
\[
(11 - 5y) + y - 2z = 5
\]
Simplifying:
\[
11 - 4y - 2z = 5
\]
\[
-4y - 2z = -6 \quad \text{so} \quad 2y + z = 3 \quad \text{(Equation D)}
\]

### Step 5: Solve the system of Equations A and D
We have:
1. \( 3y - z = 7 \quad \text{(Equation A)}\)
2. \( 2y + z = 3 \quad \text{(Equation D)}\)

Add both equations to eliminate \(z\):
\[
(3y - z) + (2y + z) = 7 + 3
\]
Simplifying:
\[
5y = 10 \quad \text{so} \quad y = 2
\]

### Step 6: Find \(z\)
Substitute \(y = 2\) into Equation D:
\[
2(2) + z = 3
\]
\[
4 + z = 3 \quad \text{so} \quad z = -1
\]

### Step 7: Find \(x\)
Substitute \(y = 2\) into Equation C:
\[
x = 11 - 5(2) = 11 - 10 = 1
\]

### Final Answer:
\[
x = 1, \quad y = 2, \quad z = -1
\]

Would you like more details on any of these steps?

Here are five related questions to explore further:
1. How can we solve this system using substitution instead of elimination?
2. How does the elimination method work with systems of equations larger than three variables?
3. Can elimination be applied to non-linear systems of equations?
4. What role do matrices play in solving systems of equations?
5. How can we check if a system of equations has no solution or infinite solutions?

**Tip:** Always double-check by substituting the solutions back into the original equations to verify correctness.

Learn how to solve a system of three linear equations using the elimination method: x+y-2z=5, -x+2y+z=2, and 2x+3y-z=9. This solution breaks down the process step-by-step, making it suitable for students in Grades 10-12.

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