Solve the following system of equations by using Matrix Inverse Method.
3x+ 4y+ 7z= 14
2x-y+ 3z= 4
2x + 2y - 3z = 0

To solve the given system of equations using the matrix inverse method, we proceed as follows:

### Step 1: Write the system of equations in matrix form.

The given system of equations is:

\[
\begin{align*}
3x + 4y + 7z &= 14 \quad \text{(Equation 1)} \\
2x - y + 3z &= 4 \quad \text{(Equation 2)} \\
2x + 2y - 3z &= 0 \quad \text{(Equation 3)}
\end{align*}
\]

This can be written in matrix form as:

\[
A \cdot X = B
\]

Where:
\[
A = \begin{pmatrix} 3 & 4 & 7 \\ 2 & -1 & 3 \\ 2 & 2 & -3 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 14 \\ 4 \\ 0 \end{pmatrix}
\]

### Step 2: Find the inverse of matrix \( A \).

To solve for \( X \), we use the matrix inverse method:

\[
X = A^{-1} \cdot B
\]

Now, I will compute the inverse of matrix \( A \) and then multiply it by matrix \( B \) to find the solution for \( x \), \( y \), and \( z \).The solution to the system of equations is:

\[
x = 0.77, \quad y = 0.94, \quad z = 1.14
\]

These values satisfy the original system of equations.

Would you like more details on the process, or do you have any further questions?

Here are 5 related questions that expand on this topic:

1. How do you find the determinant of a 3x3 matrix?
2. What are the conditions for a matrix to be invertible?
3. Can you solve a system of linear equations using Gaussian elimination?
4. How is Cramer's Rule used to solve a system of linear equations?
5. What is the significance of eigenvalues and eigenvectors in matrix algebra?

**Tip:** When solving a system of equations using the matrix inverse method, ensure the matrix is invertible by checking that its determinant is non-zero.

Learn how to solve a system of equations using the matrix inverse method. Detailed step-by-step solution to 3x + 4y + 7z = 14, 2x - y + 3z = 4, 2x + 2y - 3z = 0. Includes matrix operations and determinant calculation. Suitable for students in Grades 11-12.

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