Writing equations of lines parallel and perpendicular to a given line through a point

Question

Consider the line 

Find the equation of the line that is perpendicular to this line and passes through the point (−7,  4)

Find the equation of the line that is parallel to this line and passes through the point (−7,  4)

Answer 1

Perpendicular lines have slopes that are reciprocal of the original slope and contain the opposite sign of the original slope.

-4x - 2y = 3 lets put this into y = mx + b form

-2y = 4x + 3 now divide -2 on both sides we get

y = ( 4x/-2 ) +( 3/-2 ) = - 2x - 3/2 here the slope is just - 2

Perpendicular slope = 1/2 remember the definition of perpendicular slope from above

now lets find the equation of the Perpendicular line.

We could use point slope formula or slope intercept formula. Im going to use point slope formula:

(−7,  4) slope = 1/2

y-y1 = m(x-x1) ---> point slope formula

y - 4 = 1/2 (x - (-7))

y - 4 = 1/2x + 7/2

y = 1/2x + 7/2 + 4

y = 1/2x + 7/2 + 8/2

y = 1/2x + 15/2 perpendicular line equation

Now parallel lines contain same slope as the given line.

 (−7,  4) and m = - 2

y - 4 = - 2 (x - (-7))

y - 4 = - 2x - 14

y = -2x - 10 parallel line equation

Answer 2

-4x -2y = 3

2y = -4x -3

y = -2x -3/2 slope = -2

y-4 = -2(x+7) is the parallel line through the point (-7,4)

y-4 = (1/2)(x+7) is the perpendicular line through the point (-7, 4)

or

if you don't want point slope form

y = -2x -10 in slope intercept form for the parallel line

y= x/2 + 15/2 in slope intercept form for the perpendicular line

or if you want standard form

2x +y = -10 for the parallel line

etc.