The cheapest way -- but the most labor intensive -- is to hand weed the area and then lightly hoe the area after every rain to disturb newly germinated weeds. You might be able to recycle newspapers and then top them with partially finished compost. This is compost that is not completely ready yet. You can make your own compost for free using grass clippings and autumn leaves. You do not need a formal composter to make compost, you can just pile the materials together in a heap at least three feet tall and wide and high. I hope this gives you some ideas.
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