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

Question

Consider the line 4x+2y=-6.

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

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

Equation of Parallel line:

Equation of perpendicular line:

Answer 1

Writing equations of lines parallel and perpendicular to a given line through a point. Consider the line: 4x+2y=-6.

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

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

4x + 2y = -6 ==> 2x + y = -3 ==> y = -2x - 3 (slope: m = -2)

Equation of Parallel Line that passes through the point (4,4): m = -2, (x, y) = (4, 4)

y = mx + b ==> y = -2x + b ==> 4 = -2(4) + b ==> 4 = -8 + b ==> b = 12 ==> y = -2x + 12 ==> 2x + y = 12

Equation of perpendicular line that passes through the point (4,4): m = 1/2, (x, y) = (4, 4)

y = mx + b ==> y = (1/2)x + b ==> 4 = (1/2)(4) + b ==> 4 = 2 + b ==> b = 2 ==> y = (1/2)x + 2 ==> 2y = x + 4 ==> x - 2y = -4

Answer 2

4x+2y =-6

2y = -4x -6

y = -2x -3, slope = -2

any parallel line has the same slope

y-4 = -2(x-4) in point slope form

y = -2x +12 is the parallel line through (4,4)

a perpendicular line has a negative inverse slope.

take -2, change it's sign and flip it upside down to get +1/2

y= (1/2)x + b in slope intercept form

solve for b by plugging in x=4, y=4

4 = (1/2)4 + b

b = 2

y= x/2 + 2 is the perpendicular line through (4,4)