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The system of equations provided is:

\[
\begin{aligned}
1. & \quad 5x - 2y + 5z = 83 \\
2. & \quad 3x - 2y - 2z = 9 \\
3. & \quad x - y + 4z = 40
\end{aligned}
\]

### Step 1: Eliminate one variable (e.g., \( y \))

Subtract equation (2) from equation (1) to eliminate \( y \):

\[
(5x - 2y + 5z) - (3x - 2y - 2z) = 83 - 9
\]
\[
(5x - 3x) + (-2y + 2y) + (5z + 2z) = 74
\]
\[
2x + 7z = 74 \quad \Rightarrow \quad x + \frac{7}{2}z = 37 \quad (4)
\]

### Step 2: Eliminate \( y \) again using equation (3) and another equation

Subtract equation (3) from equation (2):

\[
(3x - 2y - 2z) - (x - y + 4z) = 9 - 40
\]
\[
(3x - x) + (-2y + y) + (-2z - 4z) = -31
\]
\[
2x - z = -31 \quad (5)
\]

### Step 3: Solve the simplified system

Now, you have the system:

\[
\begin{aligned}
x + \frac{7}{2}z & = 37 \quad (4) \\
2x - z & = -31 \quad (5)
\end{aligned}
\]

Solve equation (5) for \( z \):

\[
z = 2x + 31
\]

Substitute \( z = 2x + 31 \) into equation (4):

\[
x + \frac{7}{2}(2x + 31) = 37
\]
\[
x + 7x + \frac{7}{2}(31) = 37
\]
\[
8x + 108.5 = 37
\]
\[
8x = 37 - 108.5
\]
\[
8x = -71.5
\]
\[
x = -\frac{71.5}{8} = -8.9375
\]

### Step 4: Solve for \( z \)

Now, substitute \( x = -8.9375 \) into \( z = 2x + 31 \):

\[
z = 2(-8.9375) + 31 = -17.875 + 31 = 13.125
\]

### Step 5: Solve for \( y \)

Substitute \( x = -8.9375 \) and \( z = 13.125 \) into any original equation, for example, equation (3):

\[
x - y + 4z = 40
\]
\[
-8.9375 - y + 4(13.125) = 40
\]
\[
-8.9375 - y + 52.5 = 40
\]
\[
- y + 43.5625 = 40
\]
\[
y = 3.5625
\]

### Final Solution:
The solution set is:
\[
(x, y, z) = (-8.9375, 3.5625, 13.125)
\]

Would you like more details on any specific part?

Here are 5 related questions to expand on this:
1. What are the different methods to solve a system of linear equations?
2. How do you perform substitution and elimination more efficiently?
3. How does Gaussian elimination compare to solving systems manually?
4. What are the advantages of solving a system using matrices?
5. How can a system of equations be solved graphically?

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

Solve the system of equations:
1. 5x - 2y + 5z = 83
2. 3x - 2y - 2z = 9
3. x - y + 4z = 40

Learn how to analytically solve a system of three linear equations using substitution and elimination methods. This problem covers key algebraic concepts related to solving systems of equations. Suitable for high school students in Grades 10-12.

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