Lines (§2.3)

In the coordinate plane, the keys to understanding a line segment (which begins at one point andvends at another) are its rise and run. The key to understanding a line (which continues forever in both directions) is its slope, which is closely related to rises and runs.

Line segments

Suppose that you have two points in a rectangular coordinate system. For purposes of formulas, write their coordinates as (x₁, y₁) and (x₂, y₂); that is, x₁ is the x-coordinate of point 1, etc.

If you subtract the x-values, then the result is called the run; If you subtract the y-values, then the result is called the rise. Sometimes the symbols Δx and Δy are used for these:

run = Δx = x₂ − x₁.
rise = Δy = y₂ − y₁.

Warning: Δ is not a variable like x and y₁ are, so Δx does not mean Δ times x.

Note that the rise is positive if you move upwards on the graph while travelling from point 1 to point 2 (in addition to possibly moving horizontally as well); it's negative if you move downwards, and it's zero if you move only horizontally. Similarly, the run is positive if you move rightwards on the graph while travelling from point 1 to point 2 (in addition to possibly moving vertically as well); it's negative if you move leftwards, and it's zero if you move only vertically. In general, the rise tells you how far you move upwards, and the run tells you how for you move rightwards.

(Technically, rise and run are properties of a directed line segment, in which you travel from one endpoint to the other rather than the reverse. However, the direction makes no difference to any of the things that we'll use the rise and run for ―distance, midpoint, and slope―, as long as you're consistent about it.)

The distance between two points (or equivalently, the length of the line segment between them) can be found using the Pythagorean Theorem; the rise and run (or to be precise, their absolute values) are the lengths of the two short sides of a right triangle, and the line segment whose length we want is the long side. If d is this distance, then

Δx² + Δy² = d², so
d = √(Δx² + Δy²), or
distance = √(run² + rise²).

The midpoint is the point whose distance from each point is half their distance from each other; if its coordinates are x and y, then

(x, y) = (x₁ + ½Δx, y₁ + ½Δy), so
(x, y) = ((x₁ + x₂)/2, (y₁ + y₂)/2).

This is simply the average of the coordinates of the two points.

Lines and slopes

While a line segment starts and stops at two endpoints, a line runs forever in both directions. (Between these, a ray starts at one endpoint and then runs forever.) If you pick any two distinct points on a line, then you'll get different line segments, with different rises, runs, distances, and midpoints. However, the ratio of the rise to the run will always be the same; we call this the slope and denote it m.

slope = rise ÷ run, or
m = Δy/Δx

(If you choose the same point for both, then this gives 0/0, so choose two different points.)

The slope describes the directions in which you can travel along the line.

Lines with positive slope run up–right and down–left; lines with negative slope run down–right and up–left.
Lines whose slope has a large absolute value are steep; lines whose slope has a small absolute value are shallow.
Horizontal lines have a slope of exactly zero; vertical lines have a slope which is undefined (which you can think of as an infinite slope).

Slopes and equations

You'll want to learn this formula:

y = mx + b.

In this formula, you get the equation of a line.

The slope is the number m.
The y-intercept is (0, b).
The variables x and y stay in the equation.

If you don't know the y-intercept, you can still use this equation if you know one of the points; plug it in for x and y, and solve for b. Or use this optional formula:

y = m(x − x₁) + y₁.

Conversely, if you have an equation for the line, then solve it for y; you now know what the slope and y-intercept are. It's easy to draw a graph using any point and the slope.

Vertical lines

If a line is vertical, then the run between any two points is zero, so the slope is undefined when you divide by the run. You can also think of this as an infinite slope, since a vertical line is infinitely steep.

If a is the x-coordinate of any point, then the equation for a vertical line is always simply:

x = a.

(Similarly, if b is the y-coordinate of any point, then the equation for a horizontal line is:

y = b;

however, you don't need to learn that formula separately, since it's just the special case of y = mx + b when m = 0.)

Parallel and perpendicular lines

Since the slope of a line indicates its direction, parallel lines always have the same slope. In contrast, perpendicular lines have opposite reciprocal slopes.

Parallel lines: m₁ = m₂.
Perpendicular lines: m₁m₂ = −1, or m₂ = −1/m₁.

Also, vertical lines are parallel to one another, and horizontal and vertical lines are each perpendicular to the other. (This can't be expressed using the equations, since the slope of a vertical line is undefined.)

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