Cramer's Rule is a powerful method for solving a system of two linear equations with two unknowns. This technique utilizes determinants of matrices to find the values of the unknowns directly. To illustrate this, consider the system of equations:

1. \(2x + y = 5\)

2. \(-4x + 6y = -2\)

To apply Cramer's Rule, we first represent the coefficients and constants in an augmented matrix format:

\[\begin{bmatrix}2 & 1 & | & 5 \\-4 & 6 & | & -2\end{bmatrix}\]

In this matrix, the first column contains the coefficients of \(x\), the second column contains the coefficients of \(y\), and the third column (after the bar) contains the constants from the right side of the equations.

Cramer's Rule states that to find the value of \(x\) and \(y\), we need to calculate the determinants of specific matrices. The determinant for \(x\) is calculated by replacing the \(x\) coefficients with the constants from the right side of the equations:

\[D_x = \begin{vmatrix}5 & 1 \\-2 & 6\end{vmatrix} = (5)(6) - (1)(-2) = 30 + 2 = 32\]

The determinant for the denominator, which is the determinant of the original coefficients, is:

\[D = \begin{vmatrix}2 & 1 \\-4 & 6\end{vmatrix} = (2)(6) - (1)(-4) = 12 + 4 = 16\]

Thus, the value of \(x\) can be found using the formula:

\[x = \frac{D_x}{D} = \frac{32}{16} = 2\]

Next, we find \(y\) by replacing the \(y\) coefficients with the constants:

\[D_y = \begin{vmatrix}2 & 5 \\-4 & -2\end{vmatrix} = (2)(-2) - (5)(-4) = -4 + 20 = 16\]

Since the denominator remains the same, we can calculate \(y\) as follows:

\[y = \frac{D_y}{D} = \frac{16}{16} = 1\]

In conclusion, the solution to the system of equations is \(x = 2\) and \(y = 1\). This method not only provides a systematic approach to solving linear equations but also reinforces the concept of determinants in linear algebra.