Question

For what value of $k$ do the equations $kx-2y=3$ and $3x+y=5$ represent two lines intersecting at a unique point?

• A. $k=3$
• B. $k=-3$
• C. $k=6$
• D. all real values except $-6$

Answer 1

The correct option is D

all real values except $-6$

Step 1: Use the condition of Intersecting lines

It is given that the equation $kx-2y=3$ and $3x+y=5$ are intersecting lines.

The lines $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ are intersecting if $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$.

Here, $a_1=k,a_2=3,b_1=-2,b_2=1$. Therefore,

$\begin{array}{rcl}\frac{a_1}{a_2}& \ne & \frac{b_1}{b_2}\\ \frac{k}{3}& \ne & \frac{-2}{1}\\ k& \ne & -6\end{array}$

Hence, the required solution is that the value of $k$ is not equal to $-6$ or k is equal to all real values except $-6$.

Hence, option (d) is correct.