Investigate for what values of k & c the following system 2x + 3y = 17, kx - y = c has either no simultaneous solution or more than one simultaneous solution.
Think geometrically,for a change.
The equations correspond to equations of a line. So searching for simultaneous solutions is searching for intersections geometrically.
When will two straight lines in a plane never meet?
When will two straight lines in a plane meet at more than one point?
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Of course, you can do this using linear algebra too.
$\displaystyle \begin{vmatrix} 2 & 3 \\ k & -1 \end{vmatrix} = 0$ implies there exists "no simultaneous solution" or "more than one simultaneous solution."
So $\displaystyle k = -\frac23$
$\displaystyle kx - y = c \Rightarrow 2x+3y = -3c$
If $\displaystyle -3c = 17$,
then equations are identical and we have "more than one simultaneous solution."
else
we have "no simultaneous solution".