Given two straight lines $k x - 2 y = 5$ and $2 x + 3 y + 7 = 0$ . If the given pair of lines are perpendicular determine the value of $k$ .

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Solution

Step 1: Converting the equations into standard form of equation of a line.
The given equation of lines are $k x - 2 y = 5$ and $2 x + 3 y + 7 = 0$
Let, $m_{1,} m_{2}$ be the slope of lines $k x - 2 y = 5$ and $2 x + 3 y + 7 = 0$ respectively.
Converting the above equations in the form $y = m x + c$ .
We get,
$\Rightarrow 2 y = k x - 5$ and $3 y = - 2 x - 7$
$\Rightarrow y = \frac{k x}{2} - \frac{5}{2}$ and $y = - \frac{2 x}{3} - \frac{7}{3}$
On comparing with $y = m x + c$ , the slopes are
$m_{1} = \frac{k}{2}$ and $m_{2} = \frac{- 2}{3}$

Step 3: Determining the value of $k$ :
As it is given that , the lines are perpendicular to each other , then the multiplication of their slopes must be equal to $- 1$
$\Rightarrow m_{1} \times m_{2} = - 1$
$\Rightarrow \frac{k}{2} \times \frac{- 2}{3} = - 1$
$\Rightarrow - \frac{k}{3} = - 1$
$\Rightarrow k = 3$
Therefore , the value of $k$ is $3$ , if the lines $k x - 2 y = 5$ and $2 x + 3 y + 7 = 0$ are perpendicular.

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