With the help of determinants , solve the following systems of equations :
`2x-y=-2` , `3x+4y+8=0`

Answer

To solve the system of equations using determinants, we will follow these steps:

### Step 1: Write the equations in standard form
The given equations are:
1. \( 2x - y = -2 \)
2. \( 3x + 4y + 8 = 0 \)

We can rewrite the second equation as:
\[ 3x + 4y = -8 \]

Now, our system of equations is:
1. \( 2x - y = -2 \)
2. \( 3x + 4y = -8 \)

### Step 2: Set up the coefficient matrix and constant matrix
The coefficient matrix \( A \) and the constant matrix \( B \) can be represented as follows:
\[ A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} -2 \\ -8 \end{bmatrix} \]

### Step 3: Calculate the determinant of the coefficient matrix \( A \)
The determinant \( \Delta \) (denoted as \( |A| \)) is calculated as:
\[ \Delta = \begin{vmatrix} 2 & -1 \\ 3 & 4 \end{vmatrix} = (2 \cdot 4) - (3 \cdot -1) = 8 + 3 = 11 \]

### Step 4: Calculate \( \Delta_x \) (determinant for \( x \))
To find \( \Delta_x \), we replace the first column of \( A \) with the constant matrix \( B \):
\[ \Delta_x = \begin{vmatrix} -2 & -1 \\ -8 & 4 \end{vmatrix} = (-2 \cdot 4) - (-1 \cdot -8) = -8 - 8 = -16 \]

### Step 5: Calculate \( \Delta_y \) (determinant for \( y \))
To find \( \Delta_y \), we replace the second column of \( A \) with the constant matrix \( B \):
\[ \Delta_y = \begin{vmatrix} 2 & -2 \\ 3 & -8 \end{vmatrix} = (2 \cdot -8) - (3 \cdot -2) = -16 + 6 = -10 \]

### Step 6: Calculate the values of \( x \) and \( y \)
Using Cramer’s Rule:
\[ x = \frac{\Delta_x}{\Delta} = \frac{-16}{11}, \quad y = \frac{\Delta_y}{\Delta} = \frac{-10}{11} \]

### Final Answer
The solution to the system of equations is:
\[ x = -\frac{16}{11}, \quad y = -\frac{10}{11} \]

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