Sure! Let's consider a simple system of equations with two variables:

1. **Gaussian Elimination:** Let's solve the following system of equations using Gaussian elimination:

$$
\begin{align*}
2x + y &= 5 \\
x - y &= 1 \\
\end{align*}
$$

To solve it using Gaussian elimination, we'll perform row operations to eliminate one variable at a time. Here are the steps:

- **Step 1:** Multiply the second equation by 2 to create a cancellation term:

$$
\begin{align*}
2x + y &= 5 \\
2x - 2y &= 2 \\
\end{align*}
$$

- **Step 2:** Subtract the second equation from the first equation:

$$
\begin{align*}
2x + y &= 5 \\
-(2x - 2y) &= 2 \\
\end{align*}
$$

$$
\begin{align*}
3y &= 3 \\
\end{align*}
$$

- **Step 3:** Solve for $$ y $$:

$$
\begin{align*}
y &= 1 \\
\end{align*}
$$

- **Step 4:** Substitute the value of $$ y $$ back into one of the original equations to solve for $$ x $$:

$$
\begin{align*}
2x + 1 &= 5 \\
2x &= 4 \\
x &= 2 \\
\end{align*}
$$

Therefore, the solution to the system of equations is $$ x = 2 $$ and $$ y = 1 $$.

---

2. Cramer's Rule:
Using Cramer's rule, we can solve the same system of equations. The rule states that the solution can be found by taking the ratio of determinants. Here are the steps:

- **Step 1:** Calculate the determinant of the coefficient matrix:

$$
D = \begin{vmatrix} 2 & 1 \\ 1 & -1 \end{vmatrix} = (2 \times -1) - (1 \times 1) = -3
$$

- **Step 2:** Calculate the determinant of the $$ x $$ column (replacing the coefficients of $$ x $$ with the constants):

$$
D_x = \begin{vmatrix} 5 & 1 \\ 1 & -1 \end{vmatrix} = (5 \times -1) - (1 \times 1) = -6
$$

- **Step 3:** Calculate the determinant of the $$ y $$ column (replacing the coefficients of $$ y $$ with the constants):

$$
D_y = \begin{vmatrix} 2 & 5 \\ 1 & 1 \end{vmatrix} = (2 \times 1) - (1 \times 5) = -3
$$

- **Step 4:** Solve for $$ x $$ by dividing $$ D_x $$ by $$ D $$:

$$
x = \frac{D_x}{D} = \frac{-6}{-3} = 2
$$

- **Step 5:** Solve for $$ y $$ by dividing $$ D_y $$ by $$ D $$:

$$
y = \frac{D_y}{D} = \frac{-3}{-3} = 1
$$

Therefore, the solution to the system of equations using Cramer's rule is also $$ x = 2 $$ and $$ y = 1 $$.
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