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Strange as it may seem, we don't actually use algorithms to solve problems; an algorithm is the end-product of problem-solving. In short, every problem that has a solution already has an algorithm. Moreover, every problem that is known to have no solution has a proof to demonstrate that fact. But problems that have yet to be solved have no known algorithm or proof -- and that's precisely why they remain unsolved (for now).
There aren't. There are many problems in mathematics that have yet to be solved. If there were only five steps, they would all have been solved by now! So, obviously, there must be more than 5 steps.
Solved is the past tense of solve.
Many problems in economics can be modelled by a system of linear equations: equalities r inequalities. Such systems are best solved using matrix algebra.