One point and a slope, or two points: the same condition written twice

0:00You are given a line, but not all of it. You are given one point that lies on it, and how steeply it climbs. That is all. Now someone hands you a second point and asks whether it is on the line too. You need a rule: something to do to that point's two numbers that comes out true when it belongs and false when it does not. Here is the whole idea, and it takes one sentence.

0:27If the second point is on the line, then the steepness measured from the given point across to it is the line's own steepness. And if it is not on the line, that measured steepness is something else. So compare the two. That is the test. Draw it. The given point sits here, and I will mark it with a small one to remember it was handed to us. The point being tested floats: call its coordinates simply x and y.

1:00Join them. The steepness of that join is the rise divided by the run. The rise is y minus the given second coordinate. The run is x minus the given first. And the claim is that this quotient equals the line's slope. Write that down and you have an equation with x and y in it. Which is exactly what we were asked for. Except that it is not quite an equation about every point.

1:31Look at what sits underneath. The run is x minus the given first coordinate. If the point being tested is directly above or below the given one, that run is nothing. And you cannot divide by nothing. So there are points this rule does not answer for. It refuses. In a window of whole-numbered points six across and six up, it refuses about thirteen of them. Twelve of those thirteen are genuinely off the line, so no great loss.

2:04But one is not. One of them is the given point itself, the one point we already know is on the line. The rule cannot say so. Here is the step the derivation takes without comment. Multiply both sides by the run. On the left the run cancels and you are left with the rise. On the right you get the slope times the run. So: y minus the given second coordinate equals the slope times x minus the given first.

2:38That looks like tidying up. It is not. Put the given point into the new equation and watch. The left side is the given second coordinate minus itself, which is nothing. The right side is the slope times the given first coordinate minus itself, which is also nothing. Nothing equals nothing. It holds. The equation you multiplied out is true at a point the equation you started from could not discuss.

3:10That is worth stating plainly, because it runs against a habit. Multiplying both sides of an equation by something can change which points satisfy it. Usually that is a warning: you can gain solutions you did not want. Here you gain exactly one, and it is one you did want. Let me check that it is always the same one. Take every line in this window that has a slope and that the window sees at least three times, and take every point of it in turn as the given point.

3:46Four thousand five hundred and forty-seven combinations. In how many of them is the single point on the line that the quotient refuses to discuss the point you were given? All of them. Four thousand five hundred and forty-seven. So clearing the denominator is not a rearrangement. It is a repair, and it repairs the same hole every time. Now the half that is easy to forget. An equation of a line has to fail off the line, not merely hold on it.

4:22Suppose a point satisfies the cleared equation. Either its first coordinate differs from the given one, in which case you can divide back and its steepness from the given point really is the slope, so it is on the line. Or its first coordinate agrees, and then the right-hand side is nothing, so the left must be nothing too, so its second coordinate agrees as well. Which makes it the given point.

4:52Either way it is on the line, and there is nowhere else to be. Run that over all four and a half thousand combinations and count the points off the line the cleared equation lets in. None. And count the points of the line it misses. None. One worked case. The line through minus two, three with slope minus four. Substitute: y minus three equals minus four times x plus two.

5:23Multiply out the right and gather everything to one side. Four x plus y plus five equals nothing. Check the given point: minus eight plus three plus five. Nothing. It holds. And a point not on the line, say the origin: nothing plus nothing plus five is five, which is not nothing. It fails. Both halves, on one example. Now change what you are given. Instead of a point and a slope, you get two points and no slope at all.

5:59The picture is three points on one line: the two you were given, and the one being tested somewhere between them. Three points lie on one line exactly when the steepness between the first and the second matches the steepness between the first and the third. So write that. The steepness from the first given point to the floating point equals the steepness from the first given point to the second given point.

6:28Two quotients set equal to each other. And then, exactly as before, clear the denominators. Look at what you have. The right-hand side of that equation is a quotient built from two known points. It has a name. It is the slope. So the whole thing reads: the steepness from the given point to the floating point equals the slope. Which is the equation from ten minutes ago, character for character.

6:59The two-point form is the point-slope form with one ingredient supplied instead of handed over. It is not a second formula. It is the first one, finishing its own shopping. If you can find a slope from two points, you only ever needed to remember one of these. Take the line through one, minus one and three, five. The slope is six over two, which is three. Feeding that into point-slope gives minus three x plus y plus four equals nothing.

7:35Going through the two-point form without simplifying gives minus six x plus two y plus eight equals nothing. Those are not the same string of symbols. Every coefficient is doubled. But multiply an equation by any number that is not nothing and the points satisfying it do not move. I tried twenty different multipliers on two hundred lines. Equations that came to name a different set of points: none. So if your answer comes out as the negative of someone else's, or twice it, you have not made a mistake.

8:14You have written the same line down differently. There is a nice asymmetry hiding here. You are free to pick which point of the line you treat as the given one. For the point-slope form that freedom costs nothing at all. Whichever point of the line you start from, the equation you write down is identical. Not equivalent. Identical. The two-point form is different, because its coefficients carry the run between your two chosen points.

8:47For one line, across all the pairs the window offers, it writes as many as twenty-four different equations. Twenty-four writings, one line. And equations among them that name a different set of points: none. A related worry: does it matter which of your two points you call first? Swap them and watch both differences turn over. The rise becomes its own negative, and so does the run. A negative over a negative, so the slope survives untouched.

9:22The written equation does flip: every coefficient changes sign. Over fifteen thousand eight hundred and seventy-six ordered pairs, the number where swapping gives something other than the exact negative is none. And the number where the set of points changes at all is none. There is no correct order. Take them either way round. Both of these derivations share a hypothesis, and it is easy to walk past. The point-slope form needs a slope before it can begin.

9:58The two-point form measures one, which is the same requirement wearing a different coat. So neither can be started on a line that stands upright. This window holds thirteen upright lines. Point-slope refuses to begin on all thirteen. The two-point derivation refuses on all thirteen. That is not a technicality about division. It is why the upright family needed a section of its own before either of these could be written down.

10:32And now something the derivation does not tell you. Go back to the two-point equation after the denominators are cleared. Written out, it is a rise times a run equals a run times a rise, with no fraction anywhere. Feed it two points that stand one above the other. The run between them is nothing, so one whole term vanishes, and what is left says the first coordinate never changes.

11:01Which is exactly the upright line. Of the thirteen upright lines, the number this cleared equation names correctly is thirteen. All of them. So the blind spot was never in the equation. It was in the quotient the equation was derived through, and clearing the denominator removed it. The same repair as before, one step further out. Once you can produce a point and a slope from anything, this one equation does a great deal of work.

11:35A triangle with corners at two, one, minus two, three, and four, five. Find the segment from the last corner to the middle of the opposite side. The middle of that side is the average of its two ends: nothing across, two up. Now you have two points, so you have a slope: three quarters. Point-slope from the corner gives three x minus four y plus eight equals nothing. Or take a perpendicular. The line through two, five and minus three, six has slope minus one fifth.

12:11A right angle demands the two slopes multiply to minus one, so the one you want is five. Feed five and the point minus three, five into the same equation and you are done. Notice you never learned a formula for a median or a formula for an altitude. You learned how to produce a point and a slope, and then used the one equation you already had. Last, what this is for.

12:41A shop sells nine hundred and eighty litres a week at fourteen a litre, and one thousand two hundred and twenty litres at sixteen. Assume the relation is straight. How much at seventeen? Two points determine a line, and you already have two points. The slope is two hundred and forty over two, which is a hundred and twenty litres for each unit of price. Point-slope from the first measurement, evaluated at seventeen, gives one thousand three hundred and forty.

13:15Starting from the second measurement instead gives one thousand three hundred and forty as well. Which is the whole content of a line: it does not remember which point you built it from. One thing worth saying out loud. That slope is positive, so this law has people buying more as the price rises. That is what the numbers say. Whether it is what shops do is a different question.

13:44Two formulas, and they were one. One condition: the steepness from a known point of the line out to any other point is the line's own steepness. Whether the slope is handed to you or measured from a second point changes nothing about the condition. The step that mattered was the quiet one. Clearing the denominator is not tidying. It changes which points the equation speaks about, and it changes them in the right direction.

14:16It buys back the one point you were given, and further out it buys back a whole family of lines the derivation could never reach. An equation of a line is a claim about every point of the plane. Multiplying through was how this one came to be able to make it.