Naming a line by where it crosses the axes

0:00To write a line's equation you need a point on it and how steeply it climbs. The steepness you usually have; the point has to be given to you. Except that it does not, quite. Almost any line you draw crosses the axes, and where it crosses is a point on it, with one of its two coordinates nothing. The line is handing you a point for free. This whole idea is that, and one consequence of it that matters far more.

0:32Take the crossing of the upright axis. Every point on it has first coordinate nothing, so the crossing is nothing, and some height. Call it c. Now run the rule you already have: the height of any other point, measured from the crossing, is the steepness times how far across you have gone. y minus c equals m, times x minus nothing. And x minus nothing is x. So y equals m x plus c.

1:04Nothing new happened: a formula you already had, run at a point the picture gave away. But is that particular point allowed? Over every line this picture sees at least three times, write the point-and-steepness form at each of its points in turn. Four thousand five hundred and forty-seven equations. The number naming a different set of points from y equals m x plus c is nothing. Which point you start from never mattered. So take the free one.

1:39One thing about c is quietly mis-taught. c is not the distance from the origin up to the line. It is a coordinate, and coordinates carry a sign. Cross above the origin and c is positive; below, and it is negative. The distance is the size of c, and a distance has no sign to carry. There are six hundred and nineteen heights at which a line here can cross the upright axis, three hundred and nine of them above the origin.

2:13Every one of those has an exact mirror below: sixty-five lines cross two above the origin, and sixty-five cross two below. Same distance, different lines, and the only thing telling them apart is the sign. Nothing is special about the upright axis. The line crosses the flat one too. That crossing has second coordinate nothing, and some position along the axis. Call it d. Run the same rule there: y minus nothing equals m, times x minus d. So y is m times the bracket, x minus d.

2:55One line of working, usually left for you to do, which is a shame, because doing it is how you notice the next thing. Check it the same way. Over one thousand one hundred and ninety lines that have both, the number where the two name different sets of points is nothing. The two forms say exactly the same thing. But saying the same thing is not the same as being able to speak at all. Here is the first crack, and it is the whole video.

3:28Take a level line, running flat some way above the axis. Its steepness is nothing, and it has a height on the upright axis, so the first form is available to it. But it never crosses the flat axis. It runs alongside it forever. There is no d, and the second form cannot begin. Of the six thousand four hundred and sixty lines here, the first form can be written for six thousand four hundred and forty-seven; the second, for six thousand four hundred and thirty-four.

4:04Thirteen fewer, and those thirteen are exactly the level ones. Two forms, equally true, not equally useful. Now be greedy and take both crossings at once. The line hands you two points, and two points you can already turn into an equation. Call the crossings a along the flat axis and b up the upright one. The steepness between them is b minus nothing, over nothing minus a: minus b over a.

4:37Feed it in from the crossing at a: y equals minus b over a, times x minus a. Multiply out, gather the variables on one side, divide through by b. x over a, plus y over b, equals one. Each variable divided by its own crossing, the two pieces adding to one, and no steepness named anywhere. Look at what arrived while we were tidying. Two divisions. One by a, one by b.

5:10A division is a promise: that the thing underneath is not nothing. So this equation is not free. It carries two conditions that were not there a moment ago: the line must cross each axis somewhere other than the origin. How many lines can keep that promise? Six thousand three hundred and eighty-eight, out of six thousand four hundred and sixty. Seventy-two cannot be written this way at all, and a formula that quietly excludes anything is a formula whose shape you need to know.

5:47So which seventy-two? Guessing is what to avoid, because a guess that agrees with itself proves nothing. Find them twice, in two unrelated ways, and see whether the answers meet. First way: ask the formula. Try to build it for every line and keep the ones where it refuses. Seventy-two, found by nothing but the two divisions failing. Second way: forget the formula and describe three families out of the geometry alone.

6:20Lines through the origin: forty-eight. Level lines: thirteen. Upright lines: thirteen. Forty-eight and thirteen and thirteen is seventy-four, not seventy-two. The two extra are the axes themselves, each belonging to two families at once. So the three families together are seventy-two lines. Lines the formula refuses that lie in none of the families: nothing. Lines in a family the formula does not refuse: nothing. The same seventy-two, and neither list was built from the other.

7:00A line through the origin meets both axes there, so both crossings are nothing and both divisions have nothing underneath. A level line never reaches the flat axis, so there is no first crossing to divide by; an upright line never reaches the upright axis, so there is no second. That last pair deserves a pause, because the names invite the wrong guess. The crossing of the flat axis is the one whose second coordinate is nothing, not the one written with an x.

7:37Swap those and every answer is right up to a swap, which is the hardest kind of wrong to spot. Put the three forms side by side and something clean appears. Six thousand four hundred and sixty lines. The form from the upright crossing reaches six thousand four hundred and forty-seven, giving up only on the upright lines, which have no steepness to name. The form from the flat crossing reaches six thousand four hundred and thirty-four; the form from both crossings, six thousand three hundred and eighty-eight.

8:16A ladder, and every rung is a rung down. But here is what matters. Where two of these can both be written, do they ever disagree about which points lie on the line? Over one thousand one hundred and forty-four lines: not once. They do not differ in what they say. They differ in which lines they can say it about. The prettier the formula, the fewer lines it has.

8:44Time to use them. Two lines, both climbing one half up for every one across. The first crosses the upright axis at minus three halves: y equals x over two, minus three halves, which clears to two y minus x plus three equals nothing. The second is given by its crossing of the flat axis, at four: y equals half the bracket, x minus four, which clears to two y minus x plus four equals nothing.

9:18Identical but for a single constant. Which is what parallel looks like written down. Now the both-crossings form, on a line meeting the flat axis at minus three and the upright axis at two. Straight in: x over minus three, plus y over two, equals one. A negative crossing is nothing to fear. Minus three is a place, and the formula divides by it happily. It is nought this formula cannot survive, not a minus sign.

9:52Clear the denominators: two x minus three y plus six equals nothing. So far every line arrived as a description and left as an equation. Go the other way. An equation turns up in some shape of its own and you are asked for its steepness and its crossings. No new method: find two points satisfying it, and read the crossings off the line they make. Six x plus three y minus five equals nothing: steepness minus two, height five thirds.

10:29Three x plus two y minus twelve equals nothing: crossings at four and at six. And then this one. Three y plus two equals nothing. It has a steepness, nothing, and a height, minus two thirds. A perfectly ordinary level line. And no both-crossings form, because it never reaches the flat axis. Say that plainly rather than marking it wrong. You have been asked for a form this line does not have.

11:01Here is the same trap in nicer clothing. Find the line through the point two, three whose two crossings are equal. Equal crossings means a and b are one number, so the form reads x over a plus y over a equals one, which is x plus y equals a. Put the point in: two plus three is five, so the line is x plus y equals five, and a search of the whole field of candidates finds no other.

11:32But think about what equal crossings actually says. A line through the origin meets both axes there. Its two crossings are equal. They are both nothing. And the line through the origin and the point two, three is a real line, steepness three halves, satisfying the words of the question exactly. The search cannot see it, because the search is built from the very divisions that line breaks. Count how often two crossings agree: sixty-eight lines here, of which forty-six agree at the origin and only twenty-two anywhere else.

12:13The case that gets forgotten is the commoner one. The same shape of question, with the crossings tied together differently. Through the point two, two, with the two crossings adding to nine. Two conditions on two unknowns, so search for pairs satisfying both. Two come back: three and six, and six and three, the same numbers each way round, giving two different lines. Or tie them by a product: crossings adding to one and multiplying to minus six. Three and minus two, the only pair the search finds.

12:54Notice what all of these have in common. Not one of them hands you a steepness. You are asked to find a line entirely through where it cuts the axes, which is only possible for a line that cuts them both. One last thing, and the best argument for the y equals m x plus c shape. Take two such lines, and the upright axis, and look at the triangle the three cut out.

13:23Two of its three corners cost nothing at all. They are the two crossings of the upright axis, so the side along the axis is just the difference of the two heights. Only the third corner needs solving for, where the two lines meet, and its distance from the axis is the difference of the heights over the difference of the steepnesses. That distance is the triangle's height, so half of base times height gives the difference of the heights, squared, over twice the difference of the steepnesses.

13:59Checked against an area worked out directly from the corners: six thousand two hundred and seventy-two pairs of lines, and the number where the formula is wrong is nothing. None of the forms here is new machinery. Every one is the point-and-steepness rule run at a point the line handed over. What is new is that one of them has a domain. Divide by a crossing and you have promised, twice over, that the crossing is not nothing, and three families cannot keep that promise: the lines through the origin, the level ones and the upright ones.

14:41Seventy-two lines out of six thousand four hundred and sixty, found two independent ways that agreed exactly. So when a question asks you to put a line into a particular form, check first whether that line has one. And when it does not, the answer is not a shrug. It is which family the line belongs to, and which division would have failed.