Naming Ax + By + C = 0, the one form the distance formula ahead will take

0:00You have five ways to write a line by now, and every one of them has a line it cannot write. Four of them need a slope before they can begin, so the upright family is out. The one that uses the two crossings needs both of them to be somewhere other than the origin. Now here is a sixth, and it arrives in a single sentence. Something times x, plus something else times y, plus a number, equals nothing.

0:30With one condition attached: the two things in front of x and y are not both allowed to be nothing. That sentence is doing more work than it looks like. Two claims are hiding in it, pointing in opposite directions. Every line you have met so far can be rearranged into this shape. And every equation of this shape, if it obeys the condition, is a line. Start with the condition, because it is the strangest part.

1:02Why should the two leading coefficients not both be allowed to vanish? The usual guess is that it stops you dividing by nothing. But nothing has been divided yet. There is not a fraction anywhere in the sentence. With both of them gone, all that is left is a number equal to nothing. That is either true, or false, and it is true or false everywhere at once. I tried nine such equations across a square window of whole-numbered points.

1:35One of them was true at every single point of the window. The other eight were true at no point whatever. None of them was true at some and false at others. So the two things you get are the whole plane and the empty nothing. Neither of those is a line. That is what the condition is for. Now the first claim: every line you already know how to write fits this shape.

2:02A level line at height a: no x-term at all, then y minus a. An upright line at abscissa b: x minus b, and no y-term. A point and a slope: multiply out and gather, and you get m times x, minus y, plus the height the line started from. Two points is that same form with the slope fetched rather than handed over, so it clears the same way.

2:30A slope and a height: m times x, minus y, plus the height. And the two crossings, multiplied through by the product of them, becomes one crossing times x, plus the other times y, minus their product. So I built every one of them for all six thousand four hundred and sixty lines the window can see, and then asked each cleared equation which points it is true at. Equations that named the wrong set of points: none.

3:03Equations that came out breaking the condition: none either. But look at how far each form got. The most reaching of the five builds for six thousand four hundred and forty-seven of those lines. That leaves thirteen it cannot write at all. Every one of the thirteen stands upright. Which you would expect, because a form that needs a slope cannot start on a line that has none. The general shape builds for all six thousand four hundred and sixty.

3:37So this is not a sixth entry in a list. It is the roof over the other five. Now the other direction, which is the half people skip. Hand me three numbers, with the leading two not both nothing. Why should that describe a line? There are two cases and they are decided by one thing: whether there is a y-term to divide by. Suppose there is. Divide the whole equation through by it.

4:08You get y on its own, equal to minus the x-coefficient over the y-coefficient, times x, minus the constant over the y-coefficient. That is a slope and a height. It is the slope-and-height form, which you already know names a line. So the slope is minus A over B and the height is minus C over B. Those are claims, so I measured them. Six hundred and forty-eight triples had a y-term.

4:39Ones where minus A over B was not the slope measured from two points of the equation: none. Ones where minus C over B was not the measured height: none. Now the other case. The y-term is gone. Then the x-term cannot also be gone, because the condition forbids it. So you can divide by that one instead, and you get x equals minus the constant over the x-coefficient. One number, and x is stuck at it. That is an upright line.

5:14Seventy-two triples fell into this case. The window actually sees forty of them. Of those forty, the number whose points sat at two different abscissae: none. And the number willing to name a slope: none. Quite right, they have none. Two cases, and between them they cover everything. Seven hundred and twenty admissible triples in all. The number whose points were not exactly some line's points: none. So the fit works both ways, and the second way is the one that makes this a definition rather than a habit.

5:53There is something loose in these coefficients, and it is worth knowing about. Multiply the whole equation by any number that is not nothing. Every coefficient changes. Not one point changes. I tried seven thousand nine hundred and fifty-two multipliers on lines across the window. Ones that changed which points the equation is true at: none. Ones that broke the condition: none. For one line I counted sixteen genuinely different equations, and every one of them named that same line.

6:31The one multiplier the condition rules out is nothing itself, which flattens all three coefficients at once. Gather everything to the left, or gather everything to the right, and you get answers that are exact negatives of each other. Over nine hundred and twenty-three lines, built from their two points in both orders, the number where the second was not the exact negative of the first: none. So if your answer is the negative of somebody else's, neither of you is wrong.

7:08Which raises a question you will need. Given two of these equations, are they the same line? You cannot just compare coefficient against coefficient, because of the freedom we just found. There are three cases, and they sort themselves by ratios. If all three coefficients are in one ratio, it is the same line written twice. If only the leading two are, the lines run parallel and never meet. And if not even those are, they cross, at exactly one place.

7:43I sorted six thousand two hundred and sixteen pairs of equations this way. Forty-eight went into the same-line heap, two hundred and ninety-six into the parallel heap, and five thousand eight hundred and seventy-two into the crossing heap. Then I checked each heap against the geometry, and the number that came out wrong, in any of the three, was none. A family of parallel lines is a family sharing two coefficients and differing in the third. Hold on to that.

8:18Now the part that makes this shape worth reading rather than solving. Each coefficient going missing tells you something, and they tell you three different things. The x-coefficient gone means the line is level. The y-coefficient gone means it stands upright. The constant gone means it passes through the origin. That last one catches people out, because a missing constant feels like it should mean flat. Thirteen lines in the window have every point at one height. Thirteen lines with a missing x-coefficient. The same thirteen.

8:57Thirteen upright lines, thirteen with no y-term, the same thirteen. Forty-eight lines pass through the origin, forty-eight have no constant, the same forty-eight. In each case, lines marked that are outside the family: none. Family members the marking missed: none. Exactly one line has both the x-coefficient and the constant gone at once, and it is the flat axis, which is level and through the origin together. One equation with a letter in it. k minus three, times x. Then minus the bracket four minus k squared, times y. Then k squared, minus seven k, plus six.

9:46Read the second coefficient carefully. There is a minus sign standing in front of that bracket. So what the equation actually carries in front of y is k squared minus four, not four minus k squared. Now three questions, and all three are coefficient readings. Make it level: kill the x-coefficient. Make it upright: kill the y-coefficient. Send it through the origin: kill the constant. I tried three hundred and sixty-one values, whole numbers and fractions.

10:22Exactly one makes it level, and that is k equals three. At that value the coefficient in front of y is plus five, not minus five. Exactly two make it stand upright: two and minus two. Exactly two send it through the origin: one and six. And the number of values at which it stopped being a line at all: none. The condition never breaks here. One more thing this shape does that a slope could not.

10:55Three lines, and you want them all through one point. Two x plus y minus three. Five x plus k y minus three. Three x minus y minus two. Searching the same three hundred and sixty-one values, exactly one works: k equals minus two, and the three meet at one, one. A second set with a letter in the middle equation gives exactly one value as well, five, meeting at one, minus one.

11:27Notice what did not happen. No slope was computed anywhere. The same is true of drawing a parallel through a point you are given. Keep both leading coefficients exactly as they are, and let the point choose the constant. Six hundred and sixty point-and-line pairs. Parallels that missed the point they were built for: none. Parallels through an outside point that met the original anyway: none. So why arrange the whole thing this way?

12:01Because of what comes next: how far a point sits from a line. That distance can be written straight out of the three coefficients, and I checked it against the foot of the perpendicular, five hundred and fifty-nine times. Times the two disagreed: none. Now write the same distance with a slope in it instead. Five hundred and seven of those cases had a slope to use, and the two writings agreed every time.

12:31Fifty-two of them did not. Those are the upright lines, and the slope writing has nothing to put in. The coefficient writing answers all five hundred and fifty-nine, upright ones included. That is the reason. A distance formula written with a slope would go silent on exactly the family this shape was invented to include. Three things. The condition on the two leading coefficients is not about division. It is what stops the equation describing the whole plane, or nothing.

13:10The coefficients are not fixed by the line, only their ratios are. Two right answers can differ by a sign. And each coefficient going missing says something different: level, upright, through the origin. Underneath all of it is one fit, checked in both directions. Every line rearranges into this shape, including the ones no slope can reach. And every equation of this shape that obeys the condition is a line.

13:42That is why it can be trusted with everything that comes after it.