I ran into these type of problems occasionally, but I can never find a concrete explanation of how I'm supposed to answer a question that says to "discuss" something. this is a question I'm trying to answer:

Discuss the value of a and b such that the following system has a/ no solution, b/ unique solution or c/ infinitely many solutions. Solve case c, parametrically.

ax + y = 3

x +2y = b

exactly how do i discuss these kind of answers? help please..

  • 1
    $\begingroup$ "Discuss" is not a technical term. Rather, the exercise is asking you to find, with justification, the values for $a$ and $b$ in each case. The justification, or accompanying argument, is the "discussion". $\endgroup$ – rogerl Jun 17 '14 at 16:51
  • $\begingroup$ thanks, that helps me understand roger1. come to think of it though, im not really sure how to find a and b... $\endgroup$ – J L Jun 17 '14 at 16:54
  • $\begingroup$ or in other words: sort all pairs $(a,b)$ according to whether the system has no, $1$, or infinitely many solutions. $\endgroup$ – Greg Martin Jun 17 '14 at 17:08

To find what restrictions on $a$ and $b$ answer the three parts, you must row-reduce the augmented matrix $$\begin{pmatrix} a & 1 & 3 \\ 1 & 2 & b \end{pmatrix}$$ and then apply what you know about the relationship between row-reduced matrices and solutions of linear systems.

  • $\begingroup$ how do I row reduce this exactly? I tried switching the rows and getting rid of the 2 in the top right corner, but i couldn't get rid of the a in the bottom left... can you show me how you did it? $\endgroup$ – J L Jun 18 '14 at 15:43
  • $\begingroup$ Switch the rows, then subtract $a$ times the first row from the second, giving $\begin{pmatrix} 1 & 2 & b \\ 0 & 1-2a & 3-ab\end{pmatrix}$. Then subtract the appropriate multiple of the second row from the first. $\endgroup$ – rogerl Jun 19 '14 at 2:11

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