https://file.aiquickdraw.com/mathbot/file/1727465911144zbvg4azw.jpg

The image contains a linear system of equations presented in matrix form. The system is as follows:

\[
\begin{cases}
-3x_1 + 3x_2 - 2x_3 + 3x_4 = 6 \\
-3x_1 - 3x_2 - 3x_3 = 6 \\
x_4 = -1
\end{cases}
\]

### Step-by-Step Solution:

1. **From the third equation**, we directly have \( x_4 = -1 \).

2. **Substitute \( x_4 = -1 \) into the first and second equations:**

   - For the **first equation**:
     \[
     -3x_1 + 3x_2 - 2x_3 + 3(-1) = 6 \implies -3x_1 + 3x_2 - 2x_3 - 3 = 6 \implies -3x_1 + 3x_2 - 2x_3 = 9
     \]
     Simplifying by dividing by 3:
     \[
     -x_1 + x_2 - \frac{2}{3}x_3 = 3
     \]

   - For the **second equation**:
     \[
     -3x_1 - 3x_2 - 3x_3 = 6
     \]
     Simplifying by dividing by -3:
     \[
     x_1 + x_2 + x_3 = -2
     \]

3. **Now we have a simplified system of two equations:**

   \[
   \begin{cases}
   -x_1 + x_2 - \frac{2}{3}x_3 = 3 \\
   x_1 + x_2 + x_3 = -2
   \end{cases}
   \]

4. **Adding the two equations** to eliminate \( x_1 \):
   \[
   (-x_1 + x_2 - \frac{2}{3}x_3) + (x_1 + x_2 + x_3) = 3 - 2
   \]
   \[
   2x_2 + \frac{1}{3}x_3 = 1
   \]
   Multiply by 3 to clear the fraction:
   \[
   6x_2 + x_3 = 3 \implies x_3 = 3 - 6x_2
   \]

5. **Substitute \( x_3 = 3 - 6x_2 \) back into the second equation:**

   \[
   x_1 + x_2 + (3 - 6x_2) = -2
   \]
   \[
   x_1 + x_2 - 6x_2 + 3 = -2
   \]
   \[
   x_1 - 5x_2 = -5
   \]
   \[
   x_1 = -5 + 5x_2
   \]

### Final Solution:
The general solution in terms of the parameter \( x_2 \) is:

\[
\begin{cases}
x_1 = 5x_2 - 5 \\
x_2 = x_2 \\
x_3 = 3 - 6x_2 \\
x_4 = -1
\end{cases}
\]

### Parameter Form:
If we let \( x_2 = t \), the solution becomes:

\[
\begin{cases}
x_1 = 5t - 5 \\
x_2 = t \\
x_3 = 3 - 6t \\
x_4 = -1
\end{cases}
\]

### Do you want more details on a specific step or have any questions?

### Related Questions:
1. How can I solve linear systems using matrix operations?
2. What are the steps to convert a system into row echelon form?
3. How do parameterized solutions work in linear algebra?
4. Can we graphically interpret this solution set?
5. What are some practical applications of linear systems?

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

Solve the linear system of equations: -3x1 + 3x2 - 2x3 + 3x4 = 6, -3x1 - 3x2 - 3x3 = 6, x4 = -1.

Learn how to solve a linear system of equations using substitution and parameterization. This solution walks through each step to simplify the system, with key algebraic methods explained. Suitable for students in advanced high school or college-level linear algebra courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation