Solve the following system of equations for all three variables.
4, x, minus, 8, y, plus, 5, z, equals, minus, 8
4x−8y+5z=
−8
minus, 9, x, plus, 8, y, minus, 6, z, equals, minus, 4
−9x+8y−6z=
−4
10, x, minus, 8, y, plus, 7, z, equals, 0
10x−8y+7z=
0

To solve the given system of equations, we have:

1. \( 4x - 8y + 5z = -8 \)
2. \( -9x + 8y - 6z = -4 \)
3. \( 10x - 8y + 7z = 0 \)

We can solve this system using methods like substitution, elimination, or matrix operations. Here, I'll use the matrix method to find the values of \( x \), \( y \), and \( z \).

Let's form the coefficient matrix \( A \), the variable matrix \( X \), and the constant matrix \( B \):

\[
A = \begin{bmatrix}
4 & -8 & 5 \\
-9 & 8 & -6 \\
10 & -8 & 7
\end{bmatrix}, \quad
X = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}, \quad
B = \begin{bmatrix}
-8 \\
-4 \\
0
\end{bmatrix}
\]

The system can be written as:

\[
AX = B
\]

I will now solve this system to find the values of \( x \), \( y \), and \( z \).The solution to the system of equations is:

\[
x = 4, \quad y = -2, \quad z = -8
\]

Would you like more details on the solution process or have any other questions?

Here are 5 related questions to explore further:

1. How can we verify the solution of a system of equations?
2. What are the methods for solving a system of linear equations?
3. What is the geometric interpretation of a system of linear equations in three variables?
4. How can we determine if a system of equations has no solution or infinitely many solutions?
5. How does the determinant of a matrix relate to the solvability of a system?

**Tip:** When solving systems of linear equations, using matrix methods such as Gaussian elimination or Cramer's rule can often simplify the process.

Solve the following system of equations for all three variables: 4x - 8y + 5z = -8, -9x + 8y - 6z = -4, 10x - 8y + 7z = 0

Learn how to solve a system of three linear equations: 4x - 8y + 5z = -8, -9x + 8y - 6z = -4, and 10x - 8y + 7z = 0. This detailed solution explores matrix methods, including Gaussian elimination and Cramer's rule. Suitable for high school students in Grades 10-12.

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