Question

The slope of a line perpendicular to $5x + 3y + 1 = 0$is ________
A.$ - \dfrac{5}{3}$
B.$\dfrac{5}{3}$
C.\[ - \dfrac{3}{5}\]
D.\[\dfrac{3}{5}\]

Answer 1

Hint: Find the slope of the given line. If the equation of the line is $Ax + By + C = 0$, then the slope of the given equation is $m = - \dfrac{A}{B}$. If two lines are perpendicular to each other, then the product of the slope of the lines is equal to $ - 1$.

Complete step-by-step answer:
From the given equation of line, find the slope of the line.
If the equation of the line is $Ax + By + C = 0$, then the slope of the given equation is $m = - \dfrac{A}{B}$.
Then, the slope of the given line, $5x + 3y + 1 = 0$ is ${m_1} = - \dfrac{5}{3}$.
The product of the slopes of two perpendicular lines is $ - 1$.
Let the slope of the line perpendicular to line $5x + 3y + 1 = 0$ is ${m_2}$.
Then, we can say that, ${m_1} \times {m_2} = - 1$.
On substituting the value of ${m_1} = - \dfrac{5}{3}$, we get,
$\left( { - \dfrac{5}{3}} \right) \times {m_2} = - 1$
We will solve the above equation to find the value of ${m_2}$.
$
  \left( { - \dfrac{5}{3}} \right) \times {m_2} = - 1 \\
  {m_2} = \dfrac{3}{5} \\
 $
Therefore, the slope of line perpendicular to $5x + 3y + 1 = 0$is $\dfrac{3}{5}$.
Hence, option D is correct.

Note: If the equation of the line is $Ax + By + C = 0$, then the slope of the given equation is $m = - \dfrac{A}{B}$. If the lines are perpendicular to each other, then the product of the slope is equal to $ - 1$. If the lines are parallel to each other, then the slope of the lines are equal.