Given :-
- A equation which is 5x - 2y = 7 .
To Find :-
- The equation of the line perpendicular to the given line and passes through (3,-2) .
Solution :-
Given equation to us is ,
[tex]\longrightarrow 5x -2y = 7[/tex]
Convert it into slope intercept form which is y = mx + c ,
[tex]\longrightarrow 2y = 5x - 7 [/tex]
Divide both sides by 2 ,
[tex]\longrightarrow y =\dfrac{5}{2}x -\dfrac{7}{2} [/tex]
Now on comparing to slope intercept form , we have ,
[tex]\longrightarrow m =\dfrac{5}{2} [/tex]
And as we know that the product of slopes of two perpendicular lines is -1 . Therefore the slope of the perpendicular line will be negative reciprocal of slope of the given line . As ,
[tex]\longrightarrow m_{\perp}= \dfrac{-2}{5} [/tex]
Again the given point to us is (3,-2) . We may use the point slope form to find out the equation of perpendicular line which is ,
[tex]\longrightarrow y - y_1 = m(x-x_1)[/tex]
Substitute ,
[tex]\longrightarrow y - (-2) = \dfrac{-2}{5}(x -3)[/tex]
Open the brackets and simplify,
[tex]\longrightarrow y +2 = \dfrac{-2}{5}x +\dfrac{6}{5} [/tex]
Subtracting 2 both sides ,
[tex]\longrightarrow y=\dfrac{-2}{5}x +\dfrac{6}{5}-2 [/tex]
[tex]\longrightarrow y =\dfrac{-2}{5}x +\dfrac{6-10}{5}[/tex]
Simplify,
[tex]\longrightarrow \underline{\underline{ y = \dfrac{-2}{5}x -\dfrac{4}{5}}}[/tex]
This is the required answer !
