https://file.aiquickdraw.com/mathbot/file/1727485660965dcrzeiw8.jpg

The image contains a system of three linear equations:

\[
\begin{aligned}
2x + 1y - 3z &= 3 \\
-2y + 1z &= -4 \\
2z &= -4
\end{aligned}
\]

Let's solve this system using Gaussian elimination:

### Step 1: Start with the given system of equations

1. \( 2x + y - 3z = 3 \)
2. \( -2y + z = -4 \)
3. \( 2z = -4 \)

### Step 2: Solve the third equation for \( z \)

From the third equation \( 2z = -4 \), solving for \( z \) gives:

\[
z = \frac{-4}{2} = -2
\]

### Step 3: Substitute \( z = -2 \) into the second equation

Substitute \( z = -2 \) into the second equation \( -2y + z = -4 \):

\[
-2y + (-2) = -4 \quad \Rightarrow \quad -2y - 2 = -4
\]

Solving for \( y \):

\[
-2y = -4 + 2 \quad \Rightarrow \quad -2y = -2 \quad \Rightarrow \quad y = 1
\]

### Step 4: Substitute \( z = -2 \) and \( y = 1 \) into the first equation

Substitute \( z = -2 \) and \( y = 1 \) into the first equation \( 2x + y - 3z = 3 \):

\[
2x + 1 - 3(-2) = 3 \quad \Rightarrow \quad 2x + 1 + 6 = 3
\]

Simplifying:

\[
2x + 7 = 3 \quad \Rightarrow \quad 2x = 3 - 7 \quad \Rightarrow \quad 2x = -4
\]

Solving for \( x \):

\[
x = \frac{-4}{2} = -2
\]

### Final solution:

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

Would you like more details on this process? Here are a few related questions:

1. How do Gaussian elimination steps work for larger systems of equations?
2. Can this system be solved using matrix row reduction?
3. What are some common pitfalls in solving systems of equations with Gaussian elimination?
4. How can you check the solution to a system of equations?
5. What are the conditions for a system of equations to have no solution or infinitely many solutions?

**Tip**: Always check your solution by substituting the values back into the original equations!

Solve the following system of equations using Gaussian elimination method:
2x + 1y - 3z = 3
-2y + 1z = -4
2z = -4

Learn how to solve the linear system of equations 2x + y - 3z = 3, -2y + z = -4, 2z = -4 using the Gaussian elimination method. Step-by-step process provided, including substitution of variables and solving for x, y, and z. Suitable for students in Grades 9-12.

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