I need to convert an int variable to double

You have to cast one (or both) of the arguments to the division operator to double:

double firstSolution = (b1 * a22 - b2 * a12) / (double)(a11 * a22 - a12 * a21);

Since you are performing the same calculation twice I'd recommend refactoring your code:

double determinant = a11 * a22 - a12 * a21;
double firstSolution = (b1 * a22 - b2 * a12) / determinant;
double secondSolution = (b2 * a11 - b1 * a21) / determinant;

This works in the same way, but now there is an implicit cast to double. This conversion from int to double is an example of a widening primitive conversion.

Converting to double can be done by casting an int to a double:

You can convert an int to a double by using this mechanism like so:

int i = 3; // i is 3
double d = (double) i; // d = 3.0

Alternative (using Java's automatic type recognition):

double d = 1.0 * i; // d = 3.0

Implementing this in your code would be something like:

double firstSolution = ((double)(b1 * a22 - b2 * a12) / (double)(a11 * a22 - a12 * a21));
double secondSolution = ((double)(b2 * a11 - b1 * a21) / (double)(a11 * a22 - a12 * a21));

Alternatively you can use a hard-parameter of type double (1.0) to have java to the work for you, like so:

double firstSolution = ((1.0 * (b1 * a22 - b2 * a12)) / (1.0 * (a11 * a22 - a12 * a21)));
double secondSolution = ((1.0 * (b2 * a11 - b1 * a21)) / (1.0 * (a11 * a22 - a12 * a21)));