You make a good point if you look at that question on the surface, but I'd argue that the question can't really be answered unless we first ask, "What does it mean to eradicate poverty?" Studies have shown that people feel poor if they are poorer than the people around them. That's why a lot of people feel poor in our society even if they are living in luxury compared to someone on the other side of the globe, or even just on the other side of town. It seems to me like we are trying to increase standards of living consumeristically, and I think that's doomed to failure, plus it often makes me question "What's the point?" Because once we eradicate poverty at one level, there'll be another level to eradicate it on, even if everyone's actually living in comparative luxury. Now, I'm not arguing that poor people are living in luxury, only that I don't see the current methods being successful because even if they were, they would still feel comparatively poor and thus be subjected to the negative health outcomes associated with being under stress due to lack of control, uncertainty, etc.
What if we tried a different tack and, in addition to forms of support like rent assistance, had more widespread programs that would help people, say, develop community gardens and thus be producers (albeit only for themselves and perhaps their communities, unless they got really industrious), thereby raising their quality of living by 1) increasing their access to quality food and 2) helping them to develop a feeling of pride in what they had produced for themselves and their communities? And if people in a community produced different things, they could have engage in bartering and raise their standards of living still higher. Well, we can't do this because it wouldn't be "efficient" --- and of course because it would threaten the businesses at the top.