Using inversion method to solve the equation 2+2y +z = 20   -3x - y-z = -18   x+y +2z = 16

We are given the system of equations:

1. \( 2 + 2y + z = 20 \)
2. \( -3x - y - z = -18 \)
3. \( x + y + 2z = 16 \)

To solve this system using the **matrix inversion method**, follow these steps:

### Step 1: Convert the system of equations into matrix form

First, rewrite the equations in standard form (all variables on the left, constants on the right):

1. \( 0x + 2y + z = 18 \)  (Subtract 2 from both sides)
2. \( -3x - y - z = -18 \)
3. \( x + y + 2z = 16 \)

This system can be written as:

\[
\begin{bmatrix}
0 & 2 & 1 \\
-3 & -1 & -1 \\
1 & 1 & 2
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
18 \\
-18 \\
16
\end{bmatrix}
\]

Let:

- Matrix \( A = \begin{bmatrix} 0 & 2 & 1 \\ -3 & -1 & -1 \\ 1 & 1 & 2 \end{bmatrix} \)
- Vector \( \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \)
- Vector \( \mathbf{b} = \begin{bmatrix} 18 \\ -18 \\ 16 \end{bmatrix} \)

Thus, the system becomes \( A \mathbf{x} = \mathbf{b} \).

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

To solve for \( \mathbf{x} \), we compute \( \mathbf{x} = A^{-1} \mathbf{b} \), where \( A^{-1} \) is the inverse of matrix \( A \).

Let’s calculate the inverse of matrix \( A \).

\[
A = \begin{bmatrix}
0 & 2 & 1 \\
-3 & -1 & -1 \\
1 & 1 & 2
\end{bmatrix}
\]

To find \( A^{-1} \), we can use methods like the cofactor matrix and adjugate or use a calculator for this step. Let me compute that for you.

The solution to the system of equations is:

\[
x = 2.5, \quad y = 7.5, \quad z = 3
\]

### Summary:
Using the matrix inversion method, we found that:

- \( x = 2.5 \)
- \( y = 7.5 \)
- \( z = 3 \)

Would you like more details on any step or further clarification?

Here are 5 related questions:
1. How do you compute the determinant of a matrix?
2. What are the conditions for a matrix to have an inverse?
3. How can the Gaussian elimination method be used to solve a system of equations?
4. What is the cofactor method for finding a matrix inverse?
5. How can you verify if the solution of a system is correct?

**Tip:** A matrix is invertible (non-singular) if its determinant is non-zero.

Using inversion method to solve the equation 2 + 2y + z = 20, -3x - y - z = -18, x + y + 2z = 16

This guide demonstrates how to solve a system of linear equations using the matrix inversion method. The example system includes three equations with three variables, making it suitable for students in Grades 10-12. Learn step-by-step how to represent the system as a matrix and calculate the inverse to find the solution.

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