# what does it mean what you are asked to “discuss the value of a and b such that…”

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

• "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". – rogerl Jun 17 '14 at 16:51
• or in other words: sort all pairs $(a,b)$ according to whether the system has no, $1$, or infinitely many solutions. – 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.
• 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. – rogerl Jun 19 '14 at 2:11