3x  2y þ z ¼ 2
x  3y þ 2z ¼ 1
2x þ y  3z ¼ 5

The system of linear equations you provided is:

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

To solve this system of equations, we can use various methods, such as substitution, elimination, or matrix methods (Gaussian elimination or Cramer's rule). I'll proceed with Gaussian elimination for clarity.

### Step 1: Write the augmented matrix
We rewrite the system in matrix form \([A|b]\), where \(A\) is the coefficient matrix and \(b\) is the vector of constants on the right-hand side:

\[
\begin{bmatrix}
3 & -2 & 1 & | & 2 \\
1 & -3 & 2 & | & 1 \\
2 & 1 & -3 & | & -5
\end{bmatrix}
\]

### Step 2: Perform Gaussian elimination

1. Make the pivot in the first row, first column a 1 by swapping row 1 with row 2:

\[
\begin{bmatrix}
1 & -3 & 2 & | & 1 \\
3 & -2 & 1 & | & 2 \\
2 & 1 & -3 & | & -5
\end{bmatrix}
\]

2. Eliminate the first element in rows 2 and 3 by subtracting multiples of row 1:

- \( \text{Row 2} = \text{Row 2} - 3 \times \text{Row 1} \)
- \( \text{Row 3} = \text{Row 3} - 2 \times \text{Row 1} \)

\[
\begin{bmatrix}
1 & -3 & 2 & | & 1 \\
0 & 7 & -5 & | & -1 \\
0 & 7 & -7 & | & -7
\end{bmatrix}
\]

3. Eliminate the second element in row 3 by subtracting row 2 from row 3:

\[
\begin{bmatrix}
1 & -3 & 2 & | & 1 \\
0 & 7 & -5 & | & -1 \\
0 & 0 & -2 & | & -6
\end{bmatrix}
\]

### Step 3: Back-substitution

1. From the third equation: 
   \[
   -2z = -6 \implies z = 3
   \]
   
2. Substitute \(z = 3\) into the second equation:
   \[
   7y - 5z = -1 \implies 7y - 5(3) = -1 \implies 7y - 15 = -1 \implies y = 2
   \]
   
3. Substitute \(y = 2\) and \(z = 3\) into the first equation:
   \[
   x - 3(2) + 2(3) = 1 \implies x - 6 + 6 = 1 \implies x = 1
   \]

### Final Solution:
The solution to the system is:
\[
x = 1, \quad y = 2, \quad z = 3
\]

Would you like any more details or explanations on the steps? 

Here are some related questions for further practice:
1. Solve the system using substitution instead of Gaussian elimination.
2. Use matrix inversion to solve the system.
3. Apply Cramer's Rule to solve the system.
4. Formulate a different system of equations with three variables and solve it.
5. Use a determinant to determine if the system has a unique solution.

**Tip:** Always check your solution by substituting the values back into the original equations to verify correctness.

3x - 2y + z = 2, x - 3y + 2z = 1, 2x + y - 3z = -5

This page provides a step-by-step solution for solving the system of linear equations 3x - 2y + z = 2, x - 3y + 2z = 1, 2x + y - 3z = -5 using Gaussian elimination. Learn how to write the augmented matrix, perform row operations, and apply back-substitution to find the values of x, y, and z. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation