the distance between the lines with equations y=3/2x+4 and -3x+2y=-1

Answer 1

The two given lines are parallel, because they have the same slope. The distance between them is along a perpendicular to those two lines.

There is actually a formula for the distance between two parallel lines in the xy coordinate plane.

If the equations are in the form ax + by + c_n = 0, the formula is:

d = |c₂ - c₁| / √(a²+b²)

I. -3x + 2y - 8 = 0

II. -3x + 2y + 1 = 0

d = |- 8 - 1| / √((-3)^√2 + 2²) = 9/√13 = 9√13/13

If you did not know that formula you could take these steps:

1. Find the equation of the line through (0,4) that has a slope of -2/3 (perpendicular to both parallel lines).
2. At what point does that perpendicular line intersect the other line?
3. Use the distance formula to find the distance between (0,4) and the point discovered in 2.

y = (-2/3)x + 4 is the equation of the line perpendicular to 1st line, passing through its y-intercept.

The 2nd line in slope intercept form:

y = (3/2)x - 1/2

Using substitution:

(-2/3)x + 4 = (3/2)x - 1/2

-4x + 24 = 9x - 3

-13x = -27

x = 27/13

y = (-2/3)(27/13) + 52/13 = 34/13

What is the distance between (0,4) and (27/13, 34/13)?

√[(27/13)² + (18/13)²] = √1053 / 13 = 9√13/ 13

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