That's an interesting question. If you re-write both equations in standard form, you find:
y1(x) = -x + 5
y2(x) = -x - 1
Both equations have the same slope (-1), but different y-intercepts (5 and -1). So the vertical distance (along the y-axis) is simply 5 - (-1) = 6.
But if we're talking about about the shortest distance between them, then we need to find a line perpendicular to them both and see how far apart the intersection points are using the distance formula (Pythagorean Theorem). The perpendicular line will have a slope that is the negative reciprocal of -1 which is a new slope of -(1/-1) = 1. Just for fun we'll pick a y-intercept of 5 to match y1(x) and get a new line of:
y3(x) = x + 5
This y3(x) intersects y1(x) at (0,5). To find the intersect of y3(x) with y2(x), we set them equal to each other and solve for x.
-x - 1 = x + 5
-6 = 2x
x = -3
Subsititute for x=-3 in EITHER y3 or y2 and we get y=2, so a pair of points to look between are (0,5) and (-3,2). Find the distance between these two points, and you'll have the distance between the lines.
Sqrt((0-(-3))^2 + (5 - 2)^2) = Sqrt(3^2 + 3^2) = 3*Sqrt(2) = 4.2426...
So, the final answer(s) are, Vertical distance between the parallel lines = 6, and perpendicular distance between the parallel lines = 3*Sqrt(2) or approximately 4.2426...
