How to combine these two queries?

I have this query at the start of my index.php:

<?php query_posts($query_string . '&cat=-5'); ?>

It prevents displaying of posts from the category with the id “5” from showing in the homepage.

I also have this query:

<?php query_posts('showposts=6'); ?>

It shows only 6 posts on the homepage (a number different from the one I chose under “Blog posts show at most” in Settings >> Reading. This is because I need 20 posts to show in my archive and category pages and only 6 on the homepage.

Problem is, either one of these two tags work – not both. If I put both together everything falls apart. I don’t get 6 posts and the pagination and everything breaks. Can these two queries be combined in any way to work together?