Find the exact distance between the given lines: 2x-5y=6 2x-5y=5?

(EDIT: as "The Gnostic" has demonstrated the vertical and horizontal displacements are not 1. So my answer is wrong. I will not rewrite with the correct lengths because "The Gnostic" has done that).

These lines are parallel (quickly noticeable because the coefficients of the x and y terms are the same in each line); one lies 1 unit 'up' from the other. It also lies 1 units to the left of the other. Those vertical and horizontal displacements make a convenient right triangle with legs = 1 unit.

The altitude of this triangle is the minimal distance between the two lines and has length = (square root of 2)/2.

Please note the distance between the lines is NOT simply 1 which is the vertical displacement between them. The distance between them will have to be measured along a line that is perpendicular to both lines. This is where the right triangle and its altitude come in to the solution.

The distance between parallel lines ax+by=c and ax+by=d is |c-d|/√(a²+b²).

Here the distance is |6-5|/√(2²+(-5)²) which is 1/√29 exactly (as requested)

At x = 0, y is -1.2 on line A, and -1 on line B, so the vertical distance between the lines is .2
Ay y = -1, x is .5 on line A, and 0 on line B, so the horizontal distance between the lines is .5

The exact distance between the lines is √(horiz² + vert²) = √(.04 + .25) = 0.5385 (rounded)

Arrange lines to the form: y = m(x) + b, where m is the slope and b is the y-intercept.

2x - 5y = 6
subtract 2x to both sides:

2x [-2x] - 5y = 6 - 2x

-5y = -2x + 6
Divide both sides by -5:

-5(y) / -5 = -2(x) / -5 + 6 / -5

y = (2 / 5)(x) - 1.2

The other line:

2x - 5y = 5
Subtract 2x to both sides:

-5y = -2x + 5
Divide everything by- 5:

y = (2 / 5)(x) - 1
-----------------
Here are the two lines:

Line 1: y = (2 / 5)(x) - 1.2

Line 2: y = (2 / 5)(x) - 1
Note the lines are parallel and thus they advance at the same rate (because they have the same slope = (2 / 5)) Therefore, the distance between the lines is the difference between the y - intercepts.

| [ -1 - (-1.2)] | = 0.2 ... or | [-1.2 - (-1)] | = | -0.2| = 0.2
Absolute value signs because the distance between the 2 lines cannot be negative.