For what value of k, do the equations 

3𝑥 − 𝑦 + 8 = 0 and 6𝑥 − 𝑘𝑦 =− 16
represent coincidence lines?

                                               (OR)

If the lines given by 3𝑥 + 2𝑘𝑦 = 2 and 2𝑥 + 5𝑦 + 1 = 0 are parallel, then find the value of k.

Given the pair of equations 3𝑥 − 𝑦 + 8 = 0 and 6𝑥 − 𝑘𝑦 =− 16 represent coincidence lines.

$\begin{array}{r}3x-y+8=0\\ 6k+xy-16=0\end{array}$

For linear equations $a_{1}x+b_{1}y+c_1=0\text{ and }{a}_{2}x+b_{2}y+c_2=0\text{, if }\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$then the pair of lines will represent coincident lines

$\begin{array}{r}{a}_1=3,a_2=6,b_1=-1,b_2=-k,c_1=-8,c_2=-16\\ \Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\\ \Rightarrow \frac{3}{6}=\frac{-1}{-k}=\frac{-8}{-16}\\ \Rightarrow \frac{1}{2}=\frac{1}{k}=\frac{1}{2}\\ \Rightarrow \frac{1}{2}=\frac{1}{k}\\ \Rightarrow k=2\end{array}$Hence, the value of 𝑘 is 2.

                                              (OR)

For the pair of linear equations $a_{1}x+b_{1}y+c_1=0\text{ and }{a}_{2}x+b_{2}y+c_2=0$$\text{ If }\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$ then the pair of equations will be parallel lines. 

Here, $a_1=3,a_2=2,b_1=2k,b_2=5,c_1=-2,c_2=1$$\begin{array}{r}\text{ So, }\frac{a_1}{a_2}=\frac{3}{2}\\ \frac{b_1}{b_2}=\frac{2k}{5}\\ \frac{c_1}{c_2}=\frac{-2}{1}\\ \therefore \frac{3}{2}=\frac{2k}{5}\end{array}$

$\begin{array}{r}15=4k\\ k=\frac{15}{4}\end{array}$