Solve the following simultaneous equations using Cramer's rule:

4m + 6n = 54
3m + 2n = 28

To solve the given simultaneous equations using Cramer’s rule, follow these steps:

Equations:
1. \( 4m + 6n = 54 \)
2. \( 3m + 2n = 28 \)

### Step 1: Write the equations in standard form
Standard form:
\[ 4m + 6n = 54 \]
\[ 3m + 2n = 28 \]

### Step 2: Extract coefficients and form determinants
For Cramer's rule, define the coefficient matrix as:
\[
\text{Coefficient matrix: }
\begin{bmatrix}
4 & 6 \\
3 & 2
\end{bmatrix}
\]

Define the constant terms as:
\[
\text{Constants: }
\begin{bmatrix}
54 \\
28
\end{bmatrix}
\]

Let \( \Delta \) (the determinant of the coefficient matrix) be:
\[
\Delta = \begin{vmatrix} 4 & 6 \\ 3 & 2 \end{vmatrix} = (4 \cdot 2) - (6 \cdot 3) = 8 - 18 = -10
\]

### Step 3: Calculate determinants for \( \Delta_m \) and \( \Delta_n \)
To find \( m \) and \( n \), replace columns of the coefficient matrix with the constant matrix accordingly.

#### Determinant for \( m \) (\( \Delta_m \)):
Replace the first column with the constant matrix:
\[
\Delta_m = \begin{vmatrix} 54 & 6 \\ 28 & 2 \end{vmatrix} = (54 \cdot 2) - (6 \cdot 28) = 108 - 168 = -60
\]

#### Determinant for \( n \) (\( \Delta_n \)):
Replace the second column with the constant matrix:
\[
\Delta_n = \begin{vmatrix} 4 & 54 \\ 3 & 28 \end{vmatrix} = (4 \cdot 28) - (54 \cdot 3) = 112 - 162 = -50
\]

### Step 4: Solve for \( m \) and \( n \)
Using Cramer’s rule:
\[
m = \frac{\Delta_m}{\Delta} = \frac{-60}{-10} = 6
\]
\[
n = \frac{\Delta_n}{\Delta} = \frac{-50}{-10} = 5
\]

### Final Answer:
The solution to the simultaneous equations is:
\[
m = 6, n = 5
\]