https://file.aiquickdraw.com/mathbot/file/1725905354078f3r0p0xk.jpg

The system of linear equations from the image can be written as follows:

\[
\begin{align*}
1. & \quad x + y + z = -1 \\
2. & \quad 3x + 5y + 4z = 2 \\
3. & \quad 3x + 6y + 5z = 0
\end{align*}
\]

I will now solve this system using methods like substitution or elimination. Let's proceed.

### Step 1: Solve the first equation for one variable
From the first equation, solve for \( x \):

\[
x = -1 - y - z
\]

### Step 2: Substitute into the other two equations
Substitute \( x = -1 - y - z \) into the second and third equations.

#### Substitute into the second equation:
\[
3(-1 - y - z) + 5y + 4z = 2
\]
Simplifying:
\[
-3 - 3y - 3z + 5y + 4z = 2
\]
\[
-3 + 2y + z = 2
\]
\[
2y + z = 5 \quad \text{(Equation 4)}
\]

#### Substitute into the third equation:
\[
3(-1 - y - z) + 6y + 5z = 0
\]
Simplifying:
\[
-3 - 3y - 3z + 6y + 5z = 0
\]
\[
-3 + 3y + 2z = 0
\]
\[
3y + 2z = 3 \quad \text{(Equation 5)}
\]

### Step 3: Solve the new system
Now we have two equations with two unknowns:
1. \( 2y + z = 5 \) (Equation 4)
2. \( 3y + 2z = 3 \) (Equation 5)

Let's solve this system.

#### Multiply the first equation by 2 to eliminate \( z \):
\[
4y + 2z = 10
\]
Subtract Equation 5 from this:
\[
(4y + 2z) - (3y + 2z) = 10 - 3
\]
\[
y = 7
\]

### Step 4: Substitute \( y = 7 \) back into one of the equations
Substitute into Equation 4:
\[
2(7) + z = 5
\]
\[
14 + z = 5
\]
\[
z = -9
\]

### Step 5: Solve for \( x \)
Substitute \( y = 7 \) and \( z = -9 \) into the expression for \( x \):
\[
x = -1 - y - z = -1 - 7 - (-9) = -1 - 7 + 9 = 1
\]

### Final Solution:
\[
x = 1, \quad y = 7, \quad z = -9
\]

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

Here are some related questions to extend your understanding:
1. How do you use substitution to solve systems of linear equations?
2. What is the elimination method for solving systems of equations?
3. Can systems of equations always be solved by substitution?
4. What are some real-life applications of solving systems of equations?
5. How does the determinant of a system relate to its solvability?

**Tip:** Always check your solution by substituting the values of \( x \), \( y \), and \( z \) back into the original system of equations to verify.

Solve the system of equations: 
1. x + y + z = -1 
2. 3x + 5y + 4z = 2 
3. 3x + 6y + 5z = 0

Learn how to solve a system of linear equations with three variables using substitution and elimination methods. This step-by-step guide helps solve the equations x + y + z = -1, 3x + 5y + 4z = 2, and 3x + 6y + 5z = 0.

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