Question

For which value of k will the equations $x+2y+8=0$ and $2x+ky+14=0$ represent coincident lines?

Answer 1

The given equations are of the form
$a_{1}x+b_{1}y+c_1=0$

and $a_{2}x+b_{2}y+c_2=0$

where $a_1=1,b_1=2,c_1=7$

and $a_2=2,b_2=k,c_2=14$

The given equations will represent coincident lines if they have infinitely many solutions.

The condition for which is
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\Rightarrow \frac{1}{2}=\frac{2}{k}=\frac{7}{14}\Rightarrow k=4$

Hence, the given system of equations will represent coincident lines, if $k=4$.