Find the value of p, if the lines $y = 3 x + 7$ and $y + p x = 3$ are perpendicular to each other.

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Solution

Finding the value of $p$ :
The general equation of a straight line is given by
$y = m x + c$
where , $m =$ slope of line
If any two lines are perpendicular then the multiplication of their slopes must be equal to $- 1$ .
Given lines are $y = 3 x + 7$ and $y + p x = 3$
We are given that lines are perpendicular to each other.
${⇒m}_{1} × m_{2} = - 1$
The slope of line $y = 3 x + 7$ is $3$
The slope of line $y = - p x + 3$ is $- p$
As both the lines are given as perpendicular to each other , we can simply write
${⇒m}_{1} × m_{2} = - 1$
$\Rightarrow 3 \times (- p) = - 1 \Rightarrow p = \frac{1}{3}$
Therefore, the value of $p$ is $\frac{1}{3}$

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