You have a system of three linear equations:

\[
\begin{aligned}
1. & \quad x + y - 2z = 5 \\
2. & \quad -x + 2y + z = 2 \\
3. & \quad 2x + 3y - z = 9
\end{aligned}
\]

I will solve this system using the method of substitution, elimination, or matrix operations. Let's use the elimination method.

### Step 1: Add Equation 1 and Equation 2
Adding the first and second equations to eliminate \(x\):

\[
(x + y - 2z) + (-x + 2y + z) = 5 + 2
\]

\[
y + 2y - 2z + z = 7
\]

\[
3y - z = 7 \quad \text{(Equation 4)}
\]

### Step 2: Add Equation 2 and Equation 3
Adding the second and third equations to eliminate \(z\):

\[
(-x + 2y + z) + (2x + 3y - z) = 2 + 9
\]

\[
x + 5y = 11 \quad \text{(Equation 5)}
\]

### Step 3: Solve Equation 4 and Equation 5
Now we have:

\[
3y - z = 7 \quad \text{(Equation 4)}
\]
\[
x + 5y = 11 \quad \text{(Equation 5)}
\]

From Equation 4:

\[
z = 3y - 7
\]

Substitute this into Equation 1:

\[
x + y - 2(3y - 7) = 5
\]

Simplify and solve for \(x\):

\[
x + y - 6y + 14 = 5
\]

\[
x - 5y = -9 \quad \text{(Equation 6)}
\]

### Step 4: Solve for \(x\) and \(y\)

Now, solve the system formed by Equation 5 and Equation 6:

\[
x + 5y = 11 \quad \text{(Equation 5)}
\]
\[
x - 5y = -9 \quad \text{(Equation 6)}
\]

Add these two equations:

\[
2x = 2 \quad \Rightarrow \quad x = 1
\]

Substitute \(x = 1\) back into Equation 5:

\[
1 + 5y = 11 \quad \Rightarrow \quad 5y = 10 \quad \Rightarrow \quad y = 2
\]

### Step 5: Solve for \(z\)

Substitute \(y = 2\) into the expression for \(z\) from Equation 4:

\[
z = 3(2) - 7 = 6 - 7 = -1
\]

### Final Solution

The solution to the system is:

\[
x = 1, \quad y = 2, \quad z = -1
\]

Would you like more details, or do you have any questions?

---

Here are 5 related questions:

1. How can you verify the solution of a system of linear equations?
2. What are the methods to solve a system of three linear equations?
3. How do you represent a system of linear equations using matrices?
4. What is the determinant of a matrix, and how is it useful in solving systems of equations?
5. What are the conditions for a system of linear equations to have a unique solution?

**Tip:** Always substitute your solution back into the original equations to verify correctness.

Learn how to solve a system of three linear equations x + y - 2z = 5, -x + 2y + z = 2, 2x + 3y - z = 9 using the elimination method. Detailed step-by-step solution provided, suitable for high school students and anyone studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation