First, to change this to use a closure so that it does what you want, you need to say something like:

my $bf_1;
{ # Start a new block to avoid cluttering the environment
    my $sum_x = 0;
    $bf_1 = sub {
        # This block references $sum_x,
        # so it will be captured in the closure
        my ($operation, @args) = @_;
        # You didn't mean @ARGV; it's for command-line args
        if($operation eq 'Add') {
            $sum_x += $args[0];
        }
    };
}
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Second, your eval"" isn't going to work. $a gets set to a string containing the actual values of the various variables you reference -- including $b, which at that point is the empty string. $b will then substitute that whole string in for $a. Not really what you want. There's no magic way of simultaneously solving two equations with textual substitution and eval"".

And finally, you might want to proofread your calculations for $sum_x2 and $sum_xy. :-)