Lines parallel to an axis, where one coordinate never changes

0:00Here is a line, and here is a point somewhere on the page. The question is whether that point is on the line, and you are not allowed to look. You get one thing: the point's two coordinates, and a rule to apply to them. If the rule comes out true, the point is on the line. If it comes out false, it is off. That rule is what an equation of a line is.

0:25Notice how much that asks for. It is not enough for the rule to hold at the points of the line. It has to fail everywhere else. An equation of a line is a claim about the whole plane, not about the line. Watch what happens when you forget the second half. Take this flat line, three units above the level axis. Every point on it is at least three units up.

0:53Perfectly true, and useless. Let me test every whole-numbered point in a square window, six units out in each direction. Thirty-nine of them light up that are not on the line at all. The condition is honest about the line and silent about everything above it. In fact four different flat lines sit entirely inside that set. A rule that admits four lines has not named one. You can miss in the opposite direction too.

1:26Suppose you say: the point is the one three units up and nothing across. That is exactly one pair of numbers. It admits nothing that is off the line, so the second half is satisfied. But it misses twelve of the line's own points inside this window. So there are two ways to be wrong, and they are not the same mistake. One rule is too generous. The other is too mean.

1:53The equation of a line is the rule that is neither. Keep both halves in view, because the family we are about to meet is where the second half is easiest to lose. Now take a line drawn flat, parallel to the level axis. Walk along it and watch the two coordinates. The first one changes constantly. You can be anywhere across. The second one never moves. That is the whole description, and it is already the equation.

2:26The second coordinate equals three. Nothing about the first. Test it in both directions. Every point of the line has second coordinate three, so the rule holds on it. Every point with second coordinate three lies on it, so the rule fails off it. The absence of the first coordinate is not a gap in the equation. It is the statement that the first coordinate is free. Now I tell you only a distance: three units from the level axis.

3:01Draw the line. You cannot, and the reason matters. Distance has no sign. Three units away can be three units up or three units down. So one distance gives two lines, and they need two different equations. The second coordinate is three, or the second coordinate is minus three. This holds at every distance you can name here except one. At each of the distances one through six, exactly two flat lines answer.

3:33The exception is zero. At distance nothing the two candidates fall on top of each other and there is only one line. That one line is the level axis itself, and its equation is: the second coordinate is nothing. So the distance gives you the pair, and something else has to pick which of the two you meant. That something is position: above the axis, or below it. And now I want to be careful, because this is a place where a plausible sentence goes wrong.

4:08It is tempting to say the sign is decided by whether the line is above or below the upright axis. Look at what that would mean. The upright axis runs up the page. Of the thirteen flat lines with whole heights in this window, not one of them lies entirely to one side of it. Every single one crosses it. Nothing is above or below the upright axis, because above and below are not what that axis separates.

4:41Ask the same question of the level axis and twelve of the thirteen lie strictly to one side. The thirteenth is the axis itself. So the level axis is what decides the sign, and it has to be, because it is the one that has an above and a below. Now run the whole argument again with the two roles exchanged. A line drawn upright, parallel to the upright axis.

5:11Walk along it. The second coordinate sweeps through everything. The first one never moves. The equation is: the first coordinate equals minus two. A stated distance from the upright axis again gives two lines, one on each side. The first coordinate is that distance, or its negative. And at distance nothing the pair collapses into the upright axis, whose equation is: the first coordinate is nothing. Two families, four equations, and neither one of them needed a slope.

5:49That last remark is not a throwaway. It is the reason this whole family has to be handled separately. Almost every method for writing a line's equation starts by fixing a slope and pushing it through a point. Let me run that method exhaustively. Fifteen different slopes, every whole-numbered point in this window as the starting point, two thousand five hundred and thirty five combinations. Every flat line in the window comes out. All thirteen of them.

6:24Upright lines produced: none. Not one, and not because I chose the slopes badly. An upright line has no slope at all. There is nothing to feed in. So the two families are not equally load-bearing, and drawing them as a matched pair hides that. The flat half is a convenience, because a flat line has slope zero and the later machinery handles it. The upright half is the only route there is.

6:55Take a point, minus two across and three up. Write down the axis-parallel lines through it. There are exactly two, one from each family. The flat one holds the second coordinate at three, so its equation is: the second coordinate is three. The upright one holds the first at minus two: the first coordinate is minus two. They meet at one point, and that point is the one you started from.

7:25Every point carries exactly one line of each family, and its two coordinates are the two equations. That is really all this topic says. Which means you can read an equation and classify the line without drawing anything. Look for what is absent. If the first coordinate is missing, the first coordinate is free, and the line is flat. If the second is missing, the line is upright. If both are there, it is neither, and you will need a slope.

7:59There is one trap here worth naming. The first coordinate equals minus two is a line, not a point. It is easy to read the missing variable as zero and land on a single place. But the rule says nothing about the second coordinate, so the second coordinate may be anything. In this window that condition names thirteen points, not one. Exactly one of them has second coordinate nothing, and it is in no way special.

8:32Here is a question that looks harder than it is. Two lines cross. Find the line through their crossing that runs parallel to the upright axis. First find where they cross. Solve the pair. The crossing is at minus five twenty-seconds across and fifteen twenty-seconds up. Now write the answer. The first coordinate equals minus five twenty-seconds. That is it. Look at what you did not use. The height of the crossing point plays no part in the answer.

9:07You computed it and then threw it away. That is this topic in one line: for an upright line, only the first coordinate is doing any work. One more, and it has a sting. A right-angled triangle. The long side runs from one across, three up, to minus four across, one up. The two short sides are parallel to the axes. Find their equations. The corner has to sit where a level leg and an upright leg meet at a right angle.

9:41Search every whole-numbered point in the window for one. Two of them work, not one. The corner can be at one across and one up, giving the legs: first coordinate one, second coordinate one. Or it can be at minus four across and three up, giving: first coordinate minus four, second coordinate three. Two complete answers, and the question is phrased as though there were one. If you find one and stop, you have not finished.

10:14It is worth seeing why two appear, because it is the same double answer as before. A right angle standing on a fixed long side can go on either side of it. But that alone is not what produces exactly two here. The legs also have to run along the axes, and those two demands do different work. Watch them come apart. Stand the long side upright instead, from nothing across five up to nothing across five down.

10:46Eleven points in this window are level with both of its ends. Ten stand square on it. Points doing both: none. Same triangle, same rules, and no answer at all. So the two answers were not automatic. They were something the position of that long side happened to allow. Look back at what the two equations cost. No slope. No formula. No derivation. Just the observation that walking along one of these lines leaves one coordinate alone.

11:23And in return, a family of lines that nothing else in the subject can reach. That is the trade on offer here, and it is a good one. The equation of a line is a rule that separates the plane in two. For most lines, finding that rule takes work. For these, the rule was already written on the line, in the coordinate that never changed.