Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 12
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Question A 12 

Circle all possible solution sets for linear systems of five equations in three unknowns.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Recall the role of Gaussian Elimination in solving linear systems. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. In order to solve a linear system, we may perform Gaussian Elimination to get the row echelon form of the system's augmented matrix. If there is a zero row in the matrix representing the linear system but the righthand side is nonzero, then there is no solution. If there are more than two zero rows in the linear system matrix (i.e., the lefthand side of the augmented matrix), then it is possible for the system to have infinitely many solutions. If there are only two zero rows, and the remaining three rows of the lefthand side form an invertible matrix, then there exists a unique solution. (d) is not possible. If we assume for the sake of contradiction that the linear system has exactly two solutions and , then also solves the linear system, yielding a contradiction. Therefore, the answer is (a), (b), (c). 