Answer:
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.
