Find the condition on $k$, so that the system of equations: $x+3y=5$ and $2x+ky=8$ has a unique solution.

A system of linear equations in two variables $a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$ has a unique solution if the determinant of the coefficient matrix is non-zero:
$$\left|\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right|\ne 0⟹a_1{b}_2-a_2{b}_1\ne 0$$
Given:$1x+3y=5$$2x+ky=8$ The condition is:
$$\left|\begin{array}{cc}1& 3\\ 2& k\end{array}\right|\ne 0$$
$$(1\times k)-(2\times 3)\ne 0$$
$$k-6\ne 0⟹k\ne 6$$