To solve the system of equations using the Matrix Inverse Method, we follow these steps:

Given system of equations:

1. $3x + 4y + 7z = 14$
2. $2x - y + 3z = 4$
3. $2x + 2y - 3z = 0$

Step 1: Represent the system as a matrix equation

We can represent the system of equations in the form:

$AX = B$

Where:

• $A$ is the matrix of coefficients,
• $X$ is the column matrix of variables $[x, y, z]^T$,
• $B$ is the column matrix of constants on the right-hand side of the equations.

Coefficient matrix $A$:

3 & 4 & 7 \\ 2 & -1 & 3 \\ 2 & 2 & -3 \end{bmatrix}$$ #### Variable matrix $$X$$: $$X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$$ #### Constant matrix $$B$$: $$B = \begin{bmatrix} 14 \\ 4 \\ 0 \end{bmatrix}$$ Thus, the system becomes: $$\begin{bmatrix} 3 & 4 & 7 \\ 2 & -1 & 3 \\ 2 & 2 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 14 \\ 4 \\ 0 \end{bmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ The next step is to find the inverse of matrix $$A$$, denoted $$A^{-1}$$, and multiply it by $$B$$ to find the solution. The solution can be found using the equation: $$X = A^{-1} B$$ Let me now calculate the inverse of matrix $$A$$ and solve for $$X$$.The solution to the system of equations is: $$x = 0.765, \quad y = 0.938, \quad z = 1.136$$ Thus, the values of $$x$$, $$y$$, and $$z$$ that satisfy the system of equations are approximately $$x = 0.765$$, $$y = 0.938$$, and $$z = 1.136$$. Would you like a more detailed explanation, or do you have any questions? Here are 5 related questions: 1. How can we verify the solution to a system of equations? 2. What are the limitations of the Matrix Inverse Method? 3. Can every system of equations be solved using the Matrix Inverse Method? 4. What is the significance of the determinant of a matrix in solving systems of equations? 5. How does the Gaussian Elimination method compare to the Matrix Inverse Method? **Tip:** When the determinant of the coefficient matrix is zero, the system is either inconsistent or has infinitely many solutions, and the Matrix Inverse Method cannot be applied.