https://file.aiquickdraw.com/mathbot/file/1726374476197dp418dj0.jpg

Let's solve the system of linear equations given in the image. The equations are:

\[
1. \quad x - y + 3z = 6
\]
\[
2. \quad x - 2y = 5
\]
\[
3. \quad 2x - 2y + 5z = 9
\]

### Step 1: Solve for \( x \) from the second equation
From the second equation \( x - 2y = 5 \), we can express \( x \) in terms of \( y \):
\[
x = 5 + 2y
\]

### Step 2: Substitute into the first and third equations
Now, substitute \( x = 5 + 2y \) into the first and third equations.

#### First equation:
\[
(5 + 2y) - y + 3z = 6
\]
Simplifying:
\[
5 + y + 3z = 6
\]
\[
y + 3z = 1 \quad \text{(Equation 4)}
\]

#### Third equation:
\[
2(5 + 2y) - 2y + 5z = 9
\]
Simplifying:
\[
10 + 4y - 2y + 5z = 9
\]
\[
2y + 5z = -1 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of two equations
We now have the following two equations:
\[
4. \quad y + 3z = 1
\]
\[
5. \quad 2y + 5z = -1
\]

We can solve this system using substitution or elimination. Let's use substitution.

From equation 4, solve for \( y \):
\[
y = 1 - 3z
\]

### Step 4: Substitute into equation 5
Substitute \( y = 1 - 3z \) into equation 5:
\[
2(1 - 3z) + 5z = -1
\]
Simplify:
\[
2 - 6z + 5z = -1
\]
\[
2 - z = -1
\]
\[
z = 3
\]

### Step 5: Solve for \( y \)
Substitute \( z = 3 \) into \( y = 1 - 3z \):
\[
y = 1 - 3(3) = 1 - 9 = -8
\]

### Step 6: Solve for \( x \)
Substitute \( y = -8 \) into \( x = 5 + 2y \):
\[
x = 5 + 2(-8) = 5 - 16 = -11
\]

### Solution
The solution to the system of equations is:
\[
x = -11, \quad y = -8, \quad z = 3
\]

This is a unique solution, so the system has **one solution**.

Would you like any further details or explanations?

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Here are 5 related questions to expand on this topic:
1. How can you verify if a system of linear equations has no solution?
2. What does it mean if a system of linear equations has infinitely many solutions?
3. How can you use matrices to solve a system of equations?
4. What is the geometrical interpretation of a system of three linear equations in three variables?
5. How would the solution change if the equations were inconsistent?

**Tip**: Always check for consistency in your system of equations by using the determinant or row-reduction methods like Gaussian elimination.

Identify the number of solutions of the system of linear equations:
x - y + 3z = 6
x - 2y = 5
2x - 2y + 5z = 9

Learn how to solve the system of linear equations x - y + 3z = 6, x - 2y = 5, and 2x - 2y + 5z = 9. This problem uses substitution to find a unique solution for x, y, and z. Ideal for students in Grades 10-12.

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