https://file.aiquickdraw.com/mathbot/file/1725637450538awvcknyi.jpg

The image shows a system of three linear equations to be solved:

\[
\begin{aligned}
x - 2y + 2z &= 9 \\
-x + 3y &= -4 \\
2x - 5y + 3z &= 16
\end{aligned}
\]

We need to solve this system for the variables \(x\), \(y\), and \(z\), and express the solution as an ordered triple \((x, y, z)\).

Let me solve the system for you.

### Step 1: Add the first and second equations to eliminate \(x\).
\[
x - 2y + 2z + (-x + 3y) = 9 + (-4)
\]
This simplifies to:
\[
y + 2z = 5 \quad \text{(Equation 4)}
\]

### Step 2: Multiply the second equation by 2 and add it to the third equation to eliminate \(x\).
The second equation is:
\[
-x + 3y = -4
\]
Multiplying this by 2 gives:
\[
-2x + 6y = -8
\]
Now, add this to the third equation:
\[
(2x - 5y + 3z) + (-2x + 6y) = 16 + (-8)
\]
This simplifies to:
\[
y + 3z = 8 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of equations with \(y\) and \(z\).
We now have the two equations:
1. \(y + 2z = 5\)
2. \(y + 3z = 8\)

Subtract the first equation from the second:
\[
(y + 3z) - (y + 2z) = 8 - 5
\]
This simplifies to:
\[
z = 3
\]

### Step 4: Substitute \(z = 3\) into one of the equations (Equation 4):
\[
y + 2(3) = 5
\]
\[
y + 6 = 5
\]
\[
y = -1
\]

### Step 5: Substitute \(y = -1\) and \(z = 3\) into the first original equation:
\[
x - 2(-1) + 2(3) = 9
\]
\[
x + 2 + 6 = 9
\]
\[
x + 8 = 9
\]
\[
x = 1
\]

### Solution:
The solution to the system is:
\[
(x, y, z) = (1, -1, 3)
\]

Would you like further details or have any questions?

### Related questions:
1. How can we use matrix methods to solve systems of equations?
2. What is the substitution method for solving systems of linear equations?
3. Can systems of linear equations have no solutions or infinite solutions? How can we identify them?
4. How does the elimination method compare to substitution for solving systems of equations?
5. How can we solve larger systems of equations, such as those with four or more variables?

### Tip:
When solving a system of linear equations, always aim to eliminate one variable first to simplify the system step by step.

Solve the system: x - 2y + 2z = 9, -x + 3y = -4, 2x - 5y + 3z = 16

Discover how to solve the system of three linear equations x - 2y + 2z = 9, -x + 3y = -4, and 2x - 5y + 3z = 16 using elimination and substitution methods. This problem emphasizes key algebraic techniques suitable for students in Grades 9-12.

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