Solve the following equations using inverse of a matrix.
`{:(5x+2y=3),(3x+2y=5):}`

Answer

To solve the system of equations using the inverse of a matrix, we will follow these steps:

### Step 1: Write the equations in matrix form
We have the equations:
1. \( 5x + 2y = 3 \)
2. \( 3x + 2y = 5 \)

We can express this in matrix form as:
\[ A \cdot X = B \]
where
\[ A = \begin{pmatrix} 5 & 2 \\ 3 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 3 \\ 5 \end{pmatrix} \]

### Step 2: Find the inverse of matrix A
To find the inverse of matrix \( A \), we need to calculate the determinant of \( A \):
\[ \text{det}(A) = (5)(2) - (2)(3) = 10 - 6 = 4 \]

Since the determinant is not zero, the inverse exists. The formula for the inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

Applying this to our matrix \( A \):
\[ A^{-1} = \frac{1}{4} \begin{pmatrix} 2 & -2 \\ -3 & 5 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{2} \\ -\frac{3}{4} & \frac{5}{4} \end{pmatrix} \]

### Step 3: Multiply the inverse of A by B
Now, we will find \( X \) by multiplying \( A^{-1} \) with \( B \):
\[ X = A^{-1}B = \begin{pmatrix} \frac{1}{2} & -\frac{1}{2} \\ -\frac{3}{4} & \frac{5}{4} \end{pmatrix} \begin{pmatrix} 3 \\ 5 \end{pmatrix} \]

Calculating the multiplication:
1. For \( x \):
\[ x = \frac{1}{2} \cdot 3 + \left(-\frac{1}{2}\right) \cdot 5 = \frac{3}{2} - \frac{5}{2} = -1 \]

2. For \( y \):
\[ y = -\frac{3}{4} \cdot 3 + \frac{5}{4} \cdot 5 = -\frac{9}{4} + \frac{25}{4} = \frac{16}{4} = 4 \]

Thus, we have:
\[ X = \begin{pmatrix} -1 \\ 4 \end{pmatrix} \]

### Final Solution
The solution to the system of equations is:
\[ x = -1, \quad y = 4 \]