Equation 1: 2x + 3y = 8
Equation 2: 4x - 2y = 2

We can solve this system using both Gaussian elimination and Cramer's rule.

Gaussian Elimination:
Step 1: Write the augmented matrix for the system:

csharp
Copy code
[2   3 | 8]
[4  -2 | 2]
Step 2: Apply row operations to eliminate the coefficient below the pivot (the first element in the first row) in order to transform the matrix into reduced row-echelon form:

csharp
Copy code
[2   3 |  8]
[0  -8 | -14]
Step 3: Back-substitute to find the values of x and y:
From the second row, we can conclude that -8y = -14, so y = 14/8 = 1.75.
Substituting y = 1.75 into the first equation, we get 2x + 3(1.75) = 8.
Simplifying, 2x + 5.25 = 8, which gives us 2x = 2.75.
Finally, x = 2.75/2 = 1.375.

Therefore, the solution to the system is x = 1.375 and y = 1.75.

Cramer's Rule:
To solve the system using Cramer's rule, we need to calculate determinants.

Step 1: Calculate the determinant of the coefficient matrix (the left side of the system):
D = |2 3 |
|4 -2|

D = (2 * -2) - (3 * 4) = -14

Step 2: Calculate the determinant of the matrix obtained by replacing the first column with the constant terms:
Dx = |8 3 |
|2 -2|

Dx = (8 * -2) - (3 * 2) = -22

Step 3: Calculate the determinant of the matrix obtained by replacing the second column with the constant terms:
Dy = |2 8 |
|4 2|

Dy = (2 * 2) - (8 * 4) = -28

Step 4: Calculate the values of x and y using Cramer's rule:
x = Dx / D = -22 / -14 = 1.5714
y = Dy / D = -28 / -14 = 2

Therefore, the solution to the system using Cramer's rule is x = 1.5714 and y = 2.

Both methods provide the same solution, but Gaussian elimination offers a more efficient and straightforward approach, especially as the system becomes larger and more complex.