Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 02 (d)
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Question B 02 (d) 

In the system below, , , and are variables, and and are constants.
(d) For what value or values of and (if any) does the system have an infinite number 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! 
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Hint 

The system has infinitely many solutions if a row echelon form of the augmented matrix has more variables than nonzero rows. 
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.

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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. By (b), a row echelon form of the augmented matrix is
The system has infinitely many solutions if the row echelon form has more variables than nonzero rows. We have 3 variables . Note that first two rows cannot be zero rows because of the nonzero pivots. In order for the system to have infinitely many solutions, the third row must be a zero row. Therefore, we need and , so that the answer is . 
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Hard Easy 
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