7. A system of linear equations with fewer equations than unknowns is sometimes

Question

7. A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique solution? Explain. Choices: a. No, it cannot have a unique solution. Because there are more variables than equations, there must be at least one free variable. If there is a free variable, the solution set contains a unique solution. b. No, it cannot have a unique solution. Because there are more variables than equations, there must be at least one free variable. If the linear system is consistent and there is at least one free variable, the solution set contains infinitely many solutions. If the linear system is inconsistent, there is no solution. c. Yes, it can have a unique solution. Because there are more equations than variables, there are no free variables. If the system is consistent and there are no free variables, the solution set contains a unique solution. If the system is inconsistent, there is no solution. d. Yes, it can have a unique solution. Because there are more variables than equations, there must be at least one free variable. If the linear system is consistent and there is at least one free variable, the solution set contains either a unique solution or infinitely many solutions. If the linear system is inconsistent, there is no solution.

Explanation / Answer

Answer is No but reasoning in a) is not correct. Because free variables will give infinitely many solutions by just keeping on changing the values of free variable. We can do it because it is a free variable, it can take any value.

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