Write the system of linear equations in the form *A***x** = **b** and solve this matrix equation for **x**.

−2*x*1

 − 

3*x*2

 = 

−15

6*x*1

 + 

*x*2

 = 

−27
   
-2
   
-3
   
6
   
1

*x*1

*x*2

 = 

−15

−27




*x*1

*x*2

 = 
   
-6
   
9

To express the given system of linear equations in matrix form **A** **x** = **b**, we need to rewrite the equations and solve for **x** using matrix operations.

### Step 1: Write the system of equations.

The given system of equations is:

\[
\begin{aligned}
-2x_1 - 3x_2 &= -15 \\
6x_1 +  x_2 &= -27
\end{aligned}
\]

### Step 2: Express in matrix form **A** **x** = **b**.

We can represent the system as a matrix equation:

\[
\begin{pmatrix}
-2 & -3 \\
6 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
=
\begin{pmatrix}
-15 \\
-27
\end{pmatrix}
\]

This is in the form **A** **x** = **b**, where:

\[
\mathbf{A} =
\begin{pmatrix}
-2 & -3 \\
6 & 1
\end{pmatrix}, \quad
\mathbf{x} =
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}, \quad
\mathbf{b} =
\begin{pmatrix}
-15 \\
-27
\end{pmatrix}
\]

### Step 3: Solve for **x**.

To solve for **x**, we will find the inverse of matrix **A** (if it exists) and multiply it by **b**:

\[
\mathbf{x} = \mathbf{A}^{-1} \mathbf{b}
\]

Let's calculate this step.

I'll now compute the inverse of matrix **A** and solve the system.The solution to the system of equations is:

\[
\mathbf{x} =
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
=
\begin{pmatrix}
-6 \\
9
\end{pmatrix}
\]

Thus, \( x_1 = -6 \) and \( x_2 = 9 \).

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How do you calculate the inverse of a 2x2 matrix manually?
2. How can you check if a matrix is invertible?
3. What happens if the determinant of a matrix is zero?
4. How does matrix multiplication work in solving systems of equations?
5. What other methods can be used to solve systems of linear equations?

**Tip:** Always check if a matrix is invertible by calculating its determinant before attempting to solve using the inverse method!

Write the system of linear equations in the form A * x = b and solve this matrix equation for x.

−2x1 − 3x2 = −15
6x1 + x2 = −27

Learn how to solve a system of linear equations using matrix inversion. This example demonstrates key concepts in linear algebra, such as representing equations in matrix form and finding the inverse to solve for unknowns. Suitable for students in Grades 10-12.

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