Find the distance between the lines with equations -4x+7y+9=0 and y=4/7x-9

54/sqrt(65)

Step-by-step explanation:

You can only find the distance between two lines that are parallel.  If they are not parallel they are different lengths everywhere.  So first lets check they are parallel.  i am going to write them in slope intercept form, which is y = mx + b

-4x+7y+9=0

-4x + 9 = -7y

y = 4/7 x - 9/7

y = 4/7x - 9

the slopes (m) are the same so theya re parallel.  

Now, to find the distance we want a line that is perpendicular to the two, find where that line intersects both lines, then find the distance between those two points.  Easy right?

First a line that is perpendicular.  Super easy, a perpendicular line as a slope of -1/m.  for these two functons m = 4/7 so slope for this perpendicular equation is -7/4.

so the most simple function is y = -7/4 x so let's use that.

Now when does y = -7/4 x intersect y = 4/7 x - 9/7 and y = 4/7x - 9?  just set them equal to each other.  

-7/4 x = 4/7 x - 9/7

-7/4 x - 4/7 x = -9/7

(-49 - 16)/28 x = -9/7

-65/28 x = -9/7

x = 252/455

x = 36/65

Just double check, plug this into both equations and you should get the same answer.  

-7/4 x

-7/4 (36/65)

-63/65

4/7 x - 9/7

4/7 (36/65) - 9/7

-63/65

So the point is (36/65, -63/65)  Fun.  Next

-7/4 x =  4/7x - 9  Just gonna do this in one step.

x = 252/65

so now -7/4 252/65 = 4/7 252/65 - 9 = -441/65  So the point is (252/65, -441/65)

Now, we want the distance between  (36/65, -63/65) and  (252/65, -441/65)

sqrt((252/65 - 36/65)^2 + (-441/65 - -63/65)^2) = 54/sqrt(65)

So that's the distance between the two lines.