-- On a graph, those equations produce two lines.
-- The lines are parallel because they have the same slope ( -1 ).
-- The y-intercept of one line is zero, so it goes through the origin.
-- The y-intercept of the other line is -4 , so it crosses the y-axis at 4 units
below the origin.
-- The two lines cross the y-axis at two points that are 4 units apart vertically,
so the two lines are 4 vertical units apart everywhere.
-- If you go through the same kind of argument, you discover that they're
also 4 horizontal units apart everywhere. I think this happens because
their slopes are ' 1 ', (alright, ' -1 '), and you would find that their x-intercepts
are also ' 1 ' and ' -1 '.
-- But if you want to get technical about it . . .
You simply asked for 'the distance'. I gave you the horizontal and vertical
distances between two sloping parallel lines. A good math teacher would throw
this out. The 'distance' between parallel lines is defined as the perpendicular
distance ... the length of a line that's perpendicular to both of them.
The first method that pops into my mind goes like this:
-- The slope of these two parallel lines is -1 , so the slope of a line that's
perpendicular to them is +1 .
-- Let's look at the line that's perpendicular to them and passes through
the origin. Its equation is Y = X , and it hits one of the parallel lines there
at the origin.
-- Where does it hit the other parallel line ? Well, that line is y = -x-4 ,
and we want the point where y=x intersects it. So x = -x - 4 .
Add 'x' to each side: 2x = -4 . Divide each side by 2: x = -2 .
The point is (-2, -2) .
-- What's the length of the line between the two parallel lines that's
perpendicular to them ? We know 2 points on it . . . (0, 0) and (-2,-2) .
The distance between those points is . . .
√ [ (-2-0)² + (-2-0)² ] = √(4+4) = √8 = ( √4 ) ( √2 ) = 2 √2 .
That's about 2.8284... , and that's the real distance between those
two parallel lines.
