Capitalism is NOT a way to end poverty. It is a way to institute true justice.
Consider: in nature, a human being has to produce food and shelter in order to survive. Anything that he produces himself, by natural right belongs to him. If he works his butt off and harvests a massive field of corn, it is Right and Just that all that corn should be his. He has earned the ability to get fat that season. Conversely if he produced nothing, it is Right and Just that he have nothing. To assert otherwise is to support a contradiction.
In a free market, if you provide something of value to someone else (time, work, a product, a service... whatever), you are rewarded. The amount of your reward depends on mutual free agreement between the two people involved. If you produce nothing that anyone else cares about, you are not rewarded. This is Justice, by the same rule as the corn farmer.
In a capitalist society, if you are poor it is because you are not producing anything for anyone else. If you do anything of value for other people, you are rewarded.
Remember that business is not the only way to provide value: one of the best psychological rewards people can get is from helping other people. It feels good to help someone who you think deserves the assistance, and that good feeling is a value. When you donate to the Red Cross, you are buying that nice warm fuzzy feeling, at a free trade like anything else of value.
The problem with government solutions is that they are paid for by taxation, and you don't have a free choice about paying your taxes. So whether you support the cause or not, if government gives to it you are forced to pay for it. You are exchanging your effort for nothing, or in some cases even negative value.
For example, if you believe that evolution is an evil teaching, tough shit! The product of your effort is going to support this evil teaching, thank you government. If you don't like it, tough luck. In a private system, people only have to support the causes that THEY deem worthy of support. You can pay for a school that teaches Pastafarian Creation Theory if you like.
Another great example is the Katrina disaster. The Red Cross was there beforehand, evacuating people. They were on the scene, providing serious aid... and then the government moved in (late) and ordered them out. The government forces wouldn't let anyone else in to deliver aid, and completely botched the whole thing. People got sick and died thanks to the poor care the government gave them. Now, I chose to give to the Red Cross because in my judgment it is a well-run organization that has excellent response to disasters like Katrina. Why was my money taken to support the numbnuts who messed it up, instead? I didn't get a choice about who I thought was better for the job, I was forced to support the shitty alternative.