Question

Choose h and k such that the system below has (a) no solution, (b) a unique solution, and (c)
many solutions.

x1 + 3x2 = 2

3x1+ hx2 = k.

Answer 1

Let us consider the system

$\begin{cases} x_1 + 3x_2 = 2\\ 3x_1+ hx_2 = k \end{cases}$ (*)

Let us find the determinant $\Delta=\left|\begin{array}{cc}1 & 3 \\ 3 & h\end{array}\right|=h-9$ .

(a) If $\Delta=0$, then $h=9$, and the system (*) is equivalent to the system $\begin{cases} x_1 + 3x_2 = 2\\ x_1+ 3x_2 = \frac{k}{3} \end{cases}$ .

If $\frac{k}{3}\ne 2$, that is $k\ne 6$, then the system (*) has no solution.

(b) If $\Delta\ne 0$, that is $h\ne 9$, then for any $k$ the system (8) has a unique solution.

(c) If $\Delta=0$, that is $h=9$, and $k=6$, then the system (*) is equivalent to the system

$\begin{cases} x_1 + 3x_2 = 2\\ x_1+ 3x_2 =2 \end{cases}$ , and therefore, the system (*) has infinitely many solutions.