how do i find the distance between these lines

Question

how do i find the distance between these lines

y=2x+3

y=2x-3

Answer 1

Two lines:

2x + 3

2x - 3

Distance is shortest line between two parallel lines, which is a line perpendicular to both. Need to find points on both lines to then find distance.

Use y-intercept on 1st line as 1st point (0,3)

Find equation of line perpendicular to that line at intercept.

Slope of that line is -1/2 (negative reciprocal), so equation of line is:

y = -1/2x + b. substitute 0 for x and 3 for y

b = 3

so y = -1/2x + 3

equate that equation to the 2nd parallel line to determine point of intersection:

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

6 = 5/2 x

12/5 = x

find y by substituting value of x into equation of 2nd parallel line:

2(12/5) -3 = 9/5

2nd point is (12/5,9/5)

find distance between (0,3) and (12/5,9/5) using distance formula:

D = sqrt((12/5-0)^2 + (9/5-3)^2) = 2.68

Answer 2

The two lines are parallel. To find the distance between them you can use the formula d=|-3-3|/sqrt(2^{2}+1)=6/sqrt(5).