We are given the system of linear equations:

$5x + 2y - z = 12 \quad \text{(1)}$ $2x + 4y + z = 20 \quad \text{(2)}$ $2x + y + 2z = 15 \quad \text{(3)}$

We will solve this system using the inverse matrix method.

Step 1: Write the system in matrix form

The system of equations can be written as:

$AX = B$

Where:

• $A$ is the coefficient matrix,
• $X$ is the column vector of variables $[x, y, z]^T$,
• $B$ is the column vector of constants.

From the given equations, we extract:

5 & 2 & -1 \\ 2 & 4 & 1 \\ 2 & 1 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 12 \\ 20 \\ 15 \end{bmatrix}$$ ### Step 2: Find the inverse of matrix $$A$$ The solution to the system is given by: $$X = A^{-1}B$$ We will now calculate the inverse of matrix $$A$$. Let's compute it step by step.The solution to the system of equations is: $$x = 2, \quad y = 3, \quad z = 4$$ Thus, the values of $$x$$, $$y$$, and $$z$$ are 2, 3, and 4 respectively. Would you like further details or have any questions? Here are 5 related questions for deeper understanding: 1. How do you determine if a matrix has an inverse? 2. What is the determinant of a matrix, and why is it important in solving systems of equations? 3. Can you use the inverse matrix method for non-square systems? 4. What other methods can be used to solve simultaneous equations? 5. How does the inverse matrix method compare to Gaussian elimination? **Tip:** The inverse of a matrix exists only if its determinant is non-zero.