We are given the following system of equations:

1. \( 3x + 4y + 5z = 18 \)
2. \( 2x - y + 8z = 13 \)
3. \( 5x - 2y + 7z = 20 \)

To solve this system, we can use substitution, elimination, or matrix methods like Gaussian elimination. I'll proceed by solving it using matrix techniques (Gaussian elimination).

Let's represent this system as a matrix equation \( AX = B \):

\[
A = \begin{pmatrix} 
3 & 4 & 5 \\
2 & -1 & 8 \\
5 & -2 & 7 
\end{pmatrix}, \quad
X = \begin{pmatrix} 
x \\
y \\
z 
\end{pmatrix}, \quad
B = \begin{pmatrix} 
18 \\
13 \\
20 
\end{pmatrix}
\]

Now, I'll proceed with the Gaussian elimination method to find the values of \(x\), \(y\), and \(z\).The solution to the system of equations is:

\[
x = 3, \quad y = 1, \quad z = 1
\]

Let me know if you would like more details or have any further questions.

Here are 5 related questions you can explore:
1. How can matrix methods be applied to larger systems of equations?
2. What is the difference between Gaussian elimination and substitution methods?
3. How would the solution change if one of the equations is dependent on the others?
4. Can we solve this system using Cramer's rule?
5. How does the determinant of the matrix relate to the existence of a unique solution?

**Tip:** Matrix methods like Gaussian elimination or using inverse matrices become very powerful as the number of equations increases.

3x + 4y + 5z = 18, 2x - y + 8z = 13, 5x - 2y + 7z = 20

Learn how to solve the system of linear equations 3x + 4y + 5z = 18, 2x - y + 8z = 13, 5x - 2y + 7z = 20 using Gaussian elimination. This step-by-step guide explains matrix methods and key algebraic concepts. Suitable for high school and college students.

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