Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
x1	 − 	2x2	 + 	3x3	 = 	15
−x1	 + 	3x2	 − 	x3	 = 	−8
2x1	 − 	5x2	 + 	5x3	 = 	27


To write the given system of linear equations in the matrix form \(Ax = b\), we follow these steps:

1. **Identify the Coefficients Matrix \(A\), Variable Vector \(x\), and Constants Vector \(b\):**

   The system of equations is:

   \[
   \begin{cases}
   x_1 - 2x_2 + 3x_3 = 15 \\
   -x_1 + 3x_2 - x_3 = -8 \\
   2x_1 - 5x_2 + 5x_3 = 27
   \end{cases}
   \]

   This can be represented in matrix form as \(Ax = b\), where:

   \[
   A = \begin{bmatrix}
   1 & -2 & 3 \\
   -1 & 3 & -1 \\
   2 & -5 & 5
   \end{bmatrix},
   \quad
   x = \begin{bmatrix}
   x_1 \\
   x_2 \\
   x_3
   \end{bmatrix},
   \quad
   b = \begin{bmatrix}
   15 \\
   -8 \\
   27
   \end{bmatrix}
   \]

2. **Solve for \(x\) using the formula \(x = A^{-1}b\):**

   To solve for \(x\), we need to find the inverse of matrix \(A\) (denoted as \(A^{-1}\)) and multiply it by vector \(b\). We will calculate this step-by-step.

Let's calculate it.The solution to the system of linear equations is:

\[
x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 4 \end{bmatrix}
\]

So, the values of the variables are:
- \(x_1 = 1\)
- \(x_2 = -1\)
- \(x_3 = 4\)

Would you like more details on any specific step or have any questions?

### Related Questions:
1. How do you find the inverse of a matrix manually?
2. What are the conditions for a matrix to be invertible?
3. How can you solve a system of linear equations using row reduction?
4. What is the determinant of matrix \(A\) and why is it important?
5. Can you explain the concept of linear dependence and independence in a system of equations?

### Tip:
When solving a system of equations, always check if the determinant of the coefficient matrix is non-zero to ensure the system has a unique solution.

Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
x1 − 2x2 + 3x3 = 15
−x1 + 3x2 − x3 = −8
2x1 − 5x2 + 5x3 = 27

Learn how to solve a system of linear equations by expressing it in matrix form and using matrix inversion. This problem demonstrates key linear algebra concepts, including solving the matrix equation Ax = b and finding the inverse of a matrix. Suitable for undergraduate students.

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