https://file.aiquickdraw.com/mathbot/file/17256375836848hcjzf5i.jpg

The system of equations in the image is as follows:

\[
\begin{aligned}
1. \quad 2x + 3y - 2z &= -7 \\
2. \quad x - 2y + 4z &= 15 \\
3. \quad 2y + z &= 1
\end{aligned}
\]

Let's solve this system of equations step by step.

### Step 1: Solve for \( z \) from equation (3):
\[
2y + z = 1 \quad \Rightarrow \quad z = 1 - 2y
\]

### Step 2: Substitute \( z = 1 - 2y \) into the first and second equations.

#### Substitute into equation (1):
\[
2x + 3y - 2(1 - 2y) = -7
\]
Simplify:
\[
2x + 3y - 2 + 4y = -7 \quad \Rightarrow \quad 2x + 7y = -5 \quad \text{(Equation 4)}
\]

#### Substitute into equation (2):
\[
x - 2y + 4(1 - 2y) = 15
\]
Simplify:
\[
x - 2y + 4 - 8y = 15 \quad \Rightarrow \quad x - 10y = 11 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of two equations (Equation 4 and Equation 5):
\[
\begin{aligned}
2x + 7y &= -5 \quad \text{(Equation 4)} \\
x - 10y &= 11 \quad \text{(Equation 5)}
\end{aligned}
\]

We can solve this system by substitution or elimination. I'll use substitution by solving for \( x \) in Equation (5).

From Equation (5):
\[
x = 11 + 10y
\]

Substitute \( x = 11 + 10y \) into Equation (4):
\[
2(11 + 10y) + 7y = -5
\]
Simplify:
\[
22 + 20y + 7y = -5 \quad \Rightarrow \quad 22 + 27y = -5
\]
\[
27y = -5 - 22 = -27 \quad \Rightarrow \quad y = -1
\]

### Step 4: Substitute \( y = -1 \) back into the expression for \( x \):
\[
x = 11 + 10(-1) = 11 - 10 = 1
\]

### Step 5: Substitute \( y = -1 \) into the expression for \( z \):
\[
z = 1 - 2(-1) = 1 + 2 = 3
\]

### Final Answer:
The solution to the system is \( (x, y, z) = (1, -1, 3) \).

Would you like more details on any step? Here are five related questions for practice:

1. How would the solution change if the right-hand side of the first equation were different?
2. Can we solve this system using matrix methods like Gaussian elimination?
3. How would you verify that the solution is correct by substituting back into the original equations?
4. What happens if we express the system as a vector equation and solve graphically?
5. How would the solution be affected if one of the equations was non-linear?

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

Solve the system of linear equations provided in the image.

Learn how to solve a system of three linear equations involving three variables using substitution and elimination methods. This guide provides a step-by-step approach, explaining the reduction of variables and solving for each. Suitable for high school students studying linear algebra.

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