Why do some people believe that a mixed economic system solves basic economic problems? 

Ans) It is due to  the private sector and public sector both have a say in answering the basic economic questions, therefore, there will be a degree of high efficiency (because of the private sector involvement) and social welfare too (because of the public sector involvement).

It permits the Government to intervene when the economy faces market failure. The mixed economic system permits markets to operate freely until it fails to assign resources efficiently, after this, the government agencies, such as the EU Competition Commission, are able to correct for these failures. If this had been a totally free economic system, Government agencies would not have the means of power to be able to intervene.