The Free Market is not a solution to any problem. Obviously, a market free of regulation and red tape does not magically solve problems simply by virtue of existing. Take education, for example - the existence of an extensive voucher system would not solve any problem if nothing actually changed. So then, what makes a free-market answer worthwhile?
It allows people the ability to solve their own problems. Free-market solutions are entirely predicated on people adapting to or fixing things either by themselves or voluntarily collaborating on the basis of mutual interests in the service of common goals. It is the flexibility and freedom of a free-market market that makes it worthwhile.
Again, take education. Let's create two groups - one group consists of parents whose children are musically inclined. The other's children are athletically inclined. In a voucher system, different operators may set up a school specifically geared to each group. A music school for the students who enjoy or have aptitude for the study and performance of music, and an athletically-oriented school for the others. These schools may even co-operate, with the athletic school getting a band for it's sporting events and the music school getting a venue to perform in. Everybody wins in this market system. Each school knows exactly how much revenue they have to operate and provide services with, and each school is responsive to the needs of parents and students, as each parent can decide whether to grant the school the revenue from their child's attendance.
Whereas in today's centrally-planned school system, with it's imposed, top-down answer, both groups would be in conflict with each other, and with every other interest group in the school. As a direct result of the lack of clear lines of who-gets-what, each group tries to maximize the assets they receive from the school. School administrators don't have an incentive to entertain any of the different groups much if at all, and often make their decisions based on tradition or things that are trendy within their own circles. On top of that, many of the resources available for these programs are squandered on unnecessary staff and unproductive effort, as the school is nearly guaranteed to get a certain amount of funding every year, no matter how badly it does.
Notice how little control the people who are directly impacted have over the situation second scenario. No one in leadership has a motivation or incentive to attend to the needs of the students or parents. Instead, they're left fighting over scrap resources, or for control of the schools system itself. This is the inevitable result of top-down programs - they strip people of the ability to solve their problems, so everyone ends up fighting over who gets to impose the choice on everyone.
That is the core concept behind presenting a market as a solution - the recognition that we all do not have the same goals. Even when we do share the same general goals, we may differ on more the detailed and specific goals, or on the methods to achieve them, or even just on the order of priorities. In a free-market system, we would each be free to pursue our different goals, objectives and methods to the extent we were able. The varying degrees of success we would have would enable us to find the best methods by seeing what actually works. In the government system, "experts" far-removed from the actual problems at hand impose a single solution on everyone's problems, regardless of how badly the solution works, if the solution works at all.
Success always comes from the bottom-up, from people on the ground solving their own problems and exercising control and oversight over those that serve them. It cannot come from people who are two, three or four times removed from the problem.
A free-market allows people to solve their problems, it does not solve the problems for them.