Sure! Here is a simple example for solving systems of equations using Gaussian elimination and Cramer's rule:

Example: 
```r
2x + 3y = 8
4x - 5y = -7
```

Gaussian elimination:
```r
2   3   |   8
4  -5   |  -7

R2 = R2 - 2*R1
     2x + 3y =  8
         -11y = -23

y = 23/11
Substituting y=23/11 into 2x + 3y = 8,
x = 19/22
```

Cramer's rule:
```r
We first calculate the determinant of the coefficient matrix:

|2  3|
|4 -5| = (2*(-5)) - (3*4) = -22

Then we calculate the determinants of each matrix obtained by replacing one column of the coefficient matrix with the constant terms:
|8   3|
|-7 -5| = (8*(-5)) - (3*(-7)) = -19

|2  8|
|4 -7| = (2*(-7)) - (8*4) = -46

Finally, we obtain the solutions:
x = -19/-22 = 19/22
y = -46/-22 = 23/11
```

As for time and space complexities, Gaussian elimination has an upper bound of O(n^3) operations for an n x n system of equations, while Cramer's rule requires calculating multiple determinants, each with an upper bound of O(n^2). In terms of space complexity, both methods require storing the coefficients of the system of equations, so they are both O(n^2) in that regard. However, Gaussian elimination is an in-place algorithm, meaning that it only requires modifying the original coefficients matrix, whereas Cramer's rule requires creating several additional matrices, so it may require more memory in practice.
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