Why two points are enough to draw the line

0:00A rule like y equals two x plus one has an answer waiting for every input you can name. Nought gives one. Three gives seven. Minus fifty gives minus ninety-nine. There is no end to them, and no sheet of paper wide enough to hold them all. So drawing the rule cannot mean drawing all of it. It means drawing something that contains all of it — and the surprise is how little you have to work out to get there.

0:31Two rows of a table. That is the entire cost. Take x equals nought. Two nought is nought, plus one is one, so the first pair is nought and one. Take x equals three. Two threes are six, plus one is seven, so the second pair is three and seven. Those two inputs were chosen, not found, and both reasons are worth saying out loud. Both give whole numbers, so both land on a corner of the grid instead of somewhere between two lines.

1:06And they sit three apart, which matters more than it looks. Misplace a point by half a square with your inputs three apart and the steepness you draw is wrong by a sixth. Make exactly the same slip with them one apart and it is wrong by a half — three times as far off. The damage is always the slip divided by the span. Spread your inputs out. Mark the two points. Lay a ruler across them and draw.

1:38Then keep going, past both of them, in both directions, and put an arrow on each end. That is not decoration. The piece between the two points is the part you computed; the arrows are the part you are claiming. And the claim is a large one. At minus fifty this rule answers minus ninety-nine. At a hundred it answers two hundred and one. Nothing about the rule stops where the paper stops.

2:09Here is what the drawing quietly took on trust. Why should those two points, joined, be right about every input in between? Walk the rule instead of writing it down. Start at nought and one. Now repeat one instruction: one step across, two steps up. One and three. Two and five. Three and seven. You arrive at the second point without using the formula once. And every hop was the same hop. One across, two up. Everywhere, without exception.

2:43A path whose every step is the same movement never changes direction, and never changing direction is what straight means. Compare a path that turns by the same amount at every stage. That is just as regular and it is not straight at all — turn a quarter each time and after four steps you have drawn a square. Steady turning closes up. A steady step does not. And a machine that squares its input rises by one, then three, then five, then seven, growing every time. That growth is the bend.

3:24Now the sentence that turns a picture into mathematics. A point lies on the line exactly when its two numbers, put into the equation, make the two sides agree. Exactly when — and that phrase runs both ways. Every point on the line satisfies the equation, and every pair that satisfies the equation lies on the line. The drawing and the equation are two views of one set of pairs.

3:54Which is why the order inside a pair is not a formality. Three and seven sits on this line. Seven and three does not, because at seven the rule answers fifteen. Try the criterion on a whole row of inputs: one, two, five, seven, nine, twelve and twenty. Double and add one, each time. One gives three. Two gives five. Five gives eleven. Seven gives fifteen. Nine gives nineteen. Twelve gives twenty-five. Twenty gives forty-one.

4:30Now take one of those and test it properly. Seven and fifteen. Put seven in for x. Two sevens are fourteen, plus one is fifteen. Put fifteen in for y. Fifteen. The two sides agree, so the point is on the line. Notice what you did not do. You did not look at the drawing and decide it seemed to pass through. A pencil line has thickness. Substitution has none.

5:01Now run the whole thing backwards. Someone hands you points and no rule at all. Minus one and minus three. Nought and nought. One and three. Three and nine. Four and twelve. Set each output beside its own input and look for the relation. Minus three beside minus one. Three beside one. Nine beside three. Twelve beside four. Every output is three times its input. So the rule is y equals three x.

5:33And any two of those five points rebuild the same rule — which is the two-point fact working for you now rather than at you. One warning about how you say that, because there is a tempting wrong version. The tempting version is to divide each output by its input and watch three come out every time. And it does — at four of those five points. At nought and nought it does not. Nought divided by nought is not three. It is not anything.

6:08So say it the multiplying way rather than the dividing way. Each output is three times its input. That sentence stays true at the origin, and the dividing one falls apart there. Same fact, and only one of the two survives every point. Here is another set of points, six of them this time. Minus three and six. Minus two and four. Nought and nought. One and minus two. Two and minus four. Three and minus six.

6:43Same method, and watch it work with the signs. Six is minus two times minus three. Four is minus two times minus two. Minus six is minus two times three. So this one is y equals minus two x. Put the two rules side by side. Three x climbs three units for every step to the right. Minus two x falls two. Both pass through the origin, because neither of them adds anything on the end.

7:16The very first rule did add one, and it crosses the upright axis one unit up instead of at the corner. Between the climbing rule and the falling one, the only thing that changed was the coefficient. One more choice worth making well. Draw y equals half of x. Try x equals one and you get a half — a point sitting between two grid lines, to be guessed at with the corner of your eye.

7:46Try x equals four and you get two. A corner. So take nought and nought, and four and two, and the plotting is exact. With a coefficient of a half, the inputs that land cleanly are the even ones. With a third, they are the multiples of three. The rule for choosing is short: pick an input that the bottom of the fraction divides. Now the uncomfortable question, and it is the most important one here.

8:18Does the drawing being straight prove that the relationship is straight? It does not, and the reason is almost too simple. Any two points give a straight line. A ruler cannot help it. Watch. Take two readings off a machine that squares whatever you feed it. At nought it returns nought. At three it returns nine. Two points. Join them and you get a perfectly straight line: y equals three x. It passes through both readings exactly.

8:51Now compute a third. At one, the line says three. The machine says one. Across every input from minus forty to forty, the line and the machine agree at exactly two of them — the two you drew it from, and nowhere else. So separate the two things you have. Two points fix a line. That part is free, and it is why a two-row table is enough to draw with.

9:19Whether that line is the graph of your rule is a different question, and a ruler cannot answer it. A third pair answers it — computed from the rule, independently, and then checked against the drawing by substitution rather than by eye. Which is exactly what the table did. Every entry in it was a further test the line had to pass. Two points to draw. A third to believe it.