The distance between two parallel lines ranges from the **shortest distance** (two intersection points on a perpendicular line) to the
**horizontal distance** or **vertical distance** to an **
infinite distance**.

The **vertical distance** between the two given parallel lines is from the point (0,3) to the point (0,-3) [the two y-intercepts], which is 6. You may also compute the horizontal distance from (0,3) to (-3,3), which is 3. Now, these three points form a right triangle whose hypotenuse is divided by the perpendicular line, which intersects this hypotenuse, forming right triangles with a hypotenuse of 3 (horizontal) and a hypotenuse of 6 (vertical). This is why drawing a picture is very important.

To continue:

The line that is perpendicular to these parallel lines has a slope of (1/2) [that is, the negative reciprocal of (-2)], so the equation of that line has the form: y = (1/2)x + b and we need a point to determine b.

3 = (1/2)(0) + b

3 = b

The equation of the perpendicular line is: **y = (1/2)x + 3**

Check: The given y-intercept, (0,3), is on the perpendicular line.

The perpendicular line y=(1/2)x+3 intersects the other parallel line (y=-2x-3) when:

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

x+6 = -4x-6 [multiply both sides by 2]

5x = -12

x = -12/5

and y=(1/2)x+3

y=(1/2)(-12/5)+3 [insert value for x]

y=(-6/5)+(15/5) [common denominator]

y=(9/5)

So, the** two intersection points are (0,3) and (-12/5, 9/5).**

The distance formula is:

d = √( (x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2} )

d = √( (-12/5-0^{)2} + (9/5-3)^{2} )

d = √( 144/25 + (9/5-15/5)^{2})

d = √( 144/25 + (-6/5)^{2})

d = √(144/25 + 36/25)

d = √(180/25)

d = √(36*5/25)

** d = (6/5)√(5)**

** d = 2.68** (rounded)