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:

   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.