3x-5y+2z=-8
7x+5y-3z=16
5x+3y-7z=0               to matrix equation

To convert the given system of equations into a matrix equation, follow these steps:

1. Write the system of equations in standard form:
\[
\begin{aligned}
3x - 5y + 2z &= -8 \\
7x + 5y - 3z &= 16 \\
5x + 3y - 7z &= 0
\end{aligned}
\]

2. Express this system as a matrix equation of the form **Ax = b**, where:
   - **A** is the matrix of coefficients,
   - **x** is the vector of variables \([x, y, z]^T\),
   - **b** is the vector of constants.

The matrix form of the system is:

\[
\begin{pmatrix}
3 & -5 & 2 \\
7 & 5 & -3 \\
5 & 3 & -7
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
=
\begin{pmatrix}
-8 \\
16 \\
0
\end{pmatrix}
\]

This is the matrix equation representation of the given system of equations.

Would you like details on how to solve this system or any further explanations?

Here are 5 follow-up questions:
1. How can you solve a system of linear equations using matrix inversion?
2. What is the determinant of a 3x3 matrix, and how does it affect solving the system?
3. How do you solve a system using Cramer's Rule?
4. What are the steps to solve a system of equations using Gaussian elimination?
5. What role does matrix rank play in determining if a system has a unique solution?

**Tip:** When solving systems of equations, always check if the determinant of the coefficient matrix is non-zero. If it's zero, the system may not have a unique solution.

3x - 5y + 2z = -8, 7x + 5y - 3z = 16, 5x + 3y - 7z = 0 to matrix equation

Learn how to convert the system of linear equations 3x - 5y + 2z = -8, 7x + 5y - 3z = 16, 5x + 3y - 7z = 0 into a matrix equation. This detailed guide covers essential linear algebra concepts such as matrix representation, determinants, and solving methods like Cramer's Rule. Suitable for students in Grades 10-12.

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