Question

How do you find the slope of a line that is a)parallel and b)perpendicular to the gi $ - 2x - 5y = - 9 $ ven line ?

Answer 1

Hint: In order to determine the slope to the above equation first rewrite the equation into the slope-intercept form $ y = mx + c $ and compare with $ y = mx + c $ , m is the slope and c is the y-intercept. Slope of the line parallel will be the same as that of the given and the slope of the perpendicular will be calculated as $ - \dfrac{1}{m} $ .
Slope of line perpendicular to line having slope(m) $ = - \dfrac{1}{m} $

Complete step-by-step answer:
We are given a linear equation in two variables $ x\,and\,y $ i.e. $ - 2x - 5y = - 9 $
To determine the slope of the above equation comparing it with the slope-intercept form $ y = mx + c $
Where, m is the slope and c is the y-intercept.
Rewriting our equation
 $
   \Rightarrow - 2x - 5y = - 9 \\
   \Rightarrow - 5y = - 9 + 2x \\
   \Rightarrow y = \dfrac{{ - 9 + 2x}}{{ - 5}} \;
  $
Now ,Separating 5
\[
   \Rightarrow y = \dfrac{{ - 9}}{{ - 5}} + \dfrac{{2x}}{{( - 5)}} \\
   \Rightarrow y = \dfrac{9}{5} - \dfrac{{2x}}{5} \\
   \Rightarrow y = \left( {\dfrac{{ - 2}}{5}} \right)x + \dfrac{9}{5} \;
 \]
comparing with slope-intercept form $ y = mx + c $
So
 $
  m = \dfrac{{ - 2}}{5} \\
  c = \dfrac{9}{5} \;
  $
Hence, the slope of the equation $ m = \dfrac{{ - 2}}{5} $
So, Considering the fact that the slope of all the parallel lines to the given line always have the same slope i.e. m.
Therefore slope to line parallel to $ - 2x - 5y = - 9 $ will have slope $ m = \dfrac{{ - 2}}{5} $ .
For slope of the line perpendicular to $ - 2x - 5y = - 9 $ will be $ - \dfrac{1}{m} $
\[\dfrac{{ - 2}}{5}\]
 $
   = - \dfrac{1}{{\left( {\dfrac{{ - 2}}{5}} \right)}} \\
   = \dfrac{5}{2} \;
  $
Slope of line perpendicular = $ \dfrac{5}{2} $
Therefore the slope of line parallel and perpendicular to line $ - 2x - 5y = - 9 $ is equal to $ \dfrac{{ - 2}}{5} $ and $ \dfrac{5}{2} $ respectively.
So, the correct answer is “Therefore the slope of line parallel and perpendicular to line $ - 2x - 5y = - 9 $ is equal to $ \dfrac{{ - 2}}{5} $ and $ \dfrac{5}{2} $ respectively”.

Note: 1. Cartesian Plane: A Cartesian Plane is given its name by the French mathematician Rene Descartes ,who first used this plane in the field of mathematics .It is defined as the two mutually perpendicular number line , the one which is horizontal is given name x-axis and the one which is vertical is known as y-axis. With the help of these axes we can plot any point on this cartesian plane with the help of an ordered pair of numbers.
2.Slope-Intercept Form= $ y = mx + c $