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 parallel to this line and passes through the point (3,  -4)

Find the equation of the line that is perpendicular to this line and passes through the point (3,  -4)

Answer 1

A line that is parallel to another line will have the same slope so we have: y=-1.5X+b as our starting point. Rearranging for our unknown value of b, we see that b=y+1.5X. We know it has to pass through the point (3,-4). Now we can plug in our values and see what b is. b=-4+(1.5*3)=-4+4.5 so b=0.5. This means the line that is paralell to our line and passes through (3,-4) is y=(-3x/2)+0.5.

Perpendicular has an inverted slope so instead of -3/2, its new slope is now +2/3. We now can do the same process. y=2/3x+b, rearrange for b. b=y-(2x/3) It passes through the point (3,-4) so plug in. b=(-4)-(2*3)/3)=(-4)-2=-6. Our equation is y=2/3x-6.

Answer 2

y+4 = (-3/2)(x-3) is the parallel line through the point (3,-4) in point slope form

or

y = -3x/2 +1/2 in slope intercept form

y+4 = (2/3)(x-3) is the perpendicular line through the point (3,-4) in point slope form

or

y =2x/3 -6 in slope intercept form