Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 14
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Question A 14 

Circle all possible types of solution sets that a system of 3 equations in six unknowns can have: (a) No solutions (b) A unique solution (c) An infinite number of solutions (d) A twodimensional set of solutions (e) A fourdimensional set of solutions 
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 

We can write the system of equations in the form of matrix vector multiplication. That is, where is the vector representing the unknowns while is the corresponding matrix. 
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. Following the hint, we can simplify the question to finding the solution of where is a matrix, is a vector, and is the vector representing the unknowns. It is not possible for to be uniquely solved since has more columns than rows. If is not in the range of , there exists no solution. If is in the range of , there are infinitely many solutions. Since we have unknowns but only equations, we have freedom to assign values to at least three unknowns. Thus, the answer is . 