Classify the following system of equations as consistent or inconsistent . If consistent, solve them :
`x+y=-1`, `2x-3y=8`

Answer

To classify the given system of equations and solve them, we will follow these steps:

### Step 1: Write the system of equations
We have the following equations:
1. \( x + y = -1 \)
2. \( 2x - 3y = 8 \)

### Step 2: Write in matrix form
We can express the system in the form \( Ax = b \), where:
\[ A = \begin{pmatrix} 1 & 1 \\ 2 & -3 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \end{pmatrix}, \quad b = \begin{pmatrix} -1 \\ 8 \end{pmatrix} \]

### Step 3: Calculate the determinant of matrix A
To check the consistency of the system, we need to calculate the determinant of matrix \( A \):
\[ \text{det}(A) = (1)(-3) - (1)(2) = -3 - 2 = -5 \]

### Step 4: Check the determinant
Since \( \text{det}(A) = -5 \) (which is not equal to 0), the system of equations is consistent.

### Step 5: Find the inverse of matrix A
To solve for \( x \), we need to find the inverse of matrix \( A \):
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
The adjugate of \( A \) is obtained by swapping the diagonal elements and changing the signs of the off-diagonal elements:
\[ \text{adj}(A) = \begin{pmatrix} -3 & -1 \\ -2 & 1 \end{pmatrix} \]
Thus,
\[ A^{-1} = \frac{1}{-5} \begin{pmatrix} -3 & -1 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} \frac{3}{5} & \frac{1}{5} \\ \frac{2}{5} & -\frac{1}{5} \end{pmatrix} \]

### Step 6: Solve for \( x \)
Now we can find \( x \) using \( x = A^{-1}b \):
\[ x = A^{-1}b = \begin{pmatrix} \frac{3}{5} & \frac{1}{5} \\ \frac{2}{5} & -\frac{1}{5} \end{pmatrix} \begin{pmatrix} -1 \\ 8 \end{pmatrix} \]
Calculating this gives:
\[ x = \begin{pmatrix} \frac{3}{5} \cdot (-1) + \frac{1}{5} \cdot 8 \\ \frac{2}{5} \cdot (-1) + (-\frac{1}{5}) \cdot 8 \end{pmatrix} = \begin{pmatrix} -\frac{3}{5} + \frac{8}{5} \\ -\frac{2}{5} - \frac{8}{5} \end{pmatrix} = \begin{pmatrix} \frac{5}{5} \\ -\frac{10}{5} \end{pmatrix} = \begin{pmatrix} 1 \\ -2 \end{pmatrix} \]

### Conclusion
The solution to the system of equations is:
\[ x = 1, \quad y = -2 \]