Why a pair of these equations is a pair of straight lines

0:00Mia is at a fair with twenty coins. She rides the big wheel, and she plays ring toss. A ride on the wheel costs three coins. A game of ring toss costs four. And she plays ring toss half as many times as she rides. By the end of the afternoon she has spent all twenty coins. How many rides, and how many games? Two things we do not know, and one number — twenty — will not pin down two things by itself.

0:32The obvious thing is to try. One ride costs three coins, and half a game of ring toss is not a game at all, so that one fails before we even reach the money. Two rides cost six, one game costs four, and six and four is ten. Too little. Three rides is nine coins and a game and a half — again not whole. Four rides cost twelve, two games cost eight, and twelve and eight is exactly twenty.

1:02So four rides and two games, and we found it by trying. But notice what made that work. The answer was a small whole number, and there were only a handful of things to try. Move one price by a coin and the answer stops being whole, and trying has nowhere to go. We want a method that does not need the numbers to be kind. Start by naming what we do not know.

1:30Let x be the number of rides, and y the number of ring-toss games. Now read the story back one sentence at a time. She plays half as many games as she rides: y is a half of x. Clear the fraction and that is x minus two y equals nought. She spends twenty coins: three coins a ride, x rides, is three x; four coins a game, y games, is four y; and together they make twenty.

2:01Three x plus four y equals twenty. One sentence, one condition. Two sentences, two conditions. Neither of them, on its own, says what x is. Take just the first condition and ask what it allows. x minus two y equals nought is satisfied by x nought and y nought. It is satisfied by x two and y one. By x four and y two. By x one and y a half.

2:33There is no shortage of answers — there is an endless supply. So plot them. Every pair that satisfies the condition becomes a point, and what appears is not a scatter. It is a straight line. And you can see why it has to be. Rearrange to y equals a half x. Step one unit to the right, and y goes up by a half. Step another unit right, and y goes up by a half again.

3:03The same step, every time, wherever you start — and that is exactly what a straight line is. There is one case that needs saying separately. If the condition has no y in it at all — x equals three, say — then y can be anything, and no rearranging into y equals something will help you. Plot it anyway: every point with first coordinate three, and they stand in an upright line.

3:34So the picture is not a drawing of the equation. It is the equation's answers, all of them, laid out. To draw it you do not need many points. You need two. Put x equal to nought into x minus two y equals nought and y comes out nought. Put x equal to four and y comes out two. Two points, and exactly one line through them. A third point is worth taking, but be clear about what it is for.

4:06It is a check on your arithmetic, not on the mathematics. Two correct points have already fixed the line. Do the same for the money condition. x nought gives four y equals twenty, so y is five. x four gives twelve plus four y equals twenty, so y is two. Nought and five, four and two. It is worth being exact about what the line is, because the claim runs in two directions and only one of them is obvious.

4:40First direction: every point of the line satisfies the condition. Pick one off it anywhere — even one with fractions in it — substitute, and it works. Second direction, and this is the one a picture makes look automatic: nothing else does. Take a point just beside the line. Not far. A third of a unit off. Substitute it and the two sides no longer agree. Near the line is not on the line, and the condition does not care how near you were.

5:15So the line is exactly the answers — all of them, and nothing besides. Now put both conditions on the same sheet. The first line runs up through the origin, gaining a half for every step right. The second falls from nought and five down towards the right, because spending more on rides leaves less for games. Two lines, two lists of answers. And a pair of values that answers the whole question has to be on both lists at once.

5:48It has to satisfy the first condition, so it is somewhere on the first line. It has to satisfy the second, so it is somewhere on the second line. There is only one kind of place that is both. Where the two lines cross. That is the whole idea, and it is where a question about money becomes a question about geometry. The crossing is not a picture of the answer. It is the answer.

6:17Read it off: the lines meet where x is four and y is two. And check it against both conditions rather than one, because one is exactly what got us into this. Four minus two times two is nought. The first condition holds. Three times four plus four times two is twelve plus eight, which is twenty. The second holds. Four rides and two games. You can also reach it without drawing anything, and it is the same argument.

6:51The first condition says y is a half of x. So anywhere y appears, half of x will do instead. Put that into the money condition: three x plus four times a half x equals twenty. Four times a half x is two x, so that is three x plus two x, which is five x. Five x equals twenty, so x is four. And then y is a half of four, which is two.

7:19Same pair. And it should be — substituting the first condition into the second is doing on paper what the crossing does on the grid. One thing to be careful about, because it will matter later. We wrote the halving condition twice: once as y equals a half x, and once as x minus two y equals nought. Those look like two different equations. They are not. Multiply the first through by two and move the terms across, and you have the second.

7:54Multiplying and rearranging changes the writing. It does not change which pairs make the statement true. Take any pair you like and put it to both forms, and they agree every single time — accept together, refuse together. So it is one condition with two spellings, and it draws one line. Which also means two equations that look different can be the same line, with nothing new to tell each other.

8:25Now a warning that is easy to skip. The line does not stop where the story does. The halving line runs on through the point where x is one and y is a half. That pair genuinely satisfies the condition. But Mia cannot play half a game of ring toss. It runs backwards too, through x minus two and y minus one, and she cannot take minus two rides. So the line is the full set of answers to the equation, and the story admits only some of them.

9:00Which ones? Whole numbers, and not negative. On the money line there are exactly two such points: nought rides with five games, and four rides with two games. On the halving line there are plenty of them. But on both at once there is exactly one, and it is the answer we already have. Before we finish, one honest caution about what has just been shown. Two lines met at one point. That happened. But it is not a law, and believing it is a law will cost you later.

9:36Two straight lines on one sheet can do three things, and only three. They can cross, at exactly one point, which is what happened here. They can run parallel and never meet, and then the pair has no answer at all — two conditions that contradict each other. Or they can lie one on top of the other, and then every point of the line answers both, and there are endlessly many.

10:04Same story, different numbers. If Mia's second condition had turned out to be the halving condition rewritten, we would have learnt nothing new and been left with the whole line. So the right question is not where they cross, but what these two lines do — and crossing is one of three answers. That last point is not a caution added for politeness. It is the reason the checking here was built the way it was.

10:34'Two lines cross at one point' is true of this pair — so a machine that could only ever answer 'one point' would agree with every word of this and prove nothing whatever. So the machine used here was handed nine thousand seven hundred and thirty pairs of conditions, including parallel ones and repeated ones, and it had to say which of the three each pair was. It answered 'one point' for eight thousand eight hundred and eighty, 'no answer' for eight hundred and nine, and 'the whole line' for forty-one.

11:10It could say all three, so 'one point' meant something when it said it here. The same worry applies to the line itself. Reading points off a line and then checking they are on it confirms nothing. So the points were found by searching — walking a grid and keeping the pairs that pass — and only then was the line drawn through the first two of them. Every one of the others landed on it — three thousand five hundred and seventy-five of them.

11:42And each was then pushed a third of a unit sideways, across the line rather than along it, and every single one stopped working. Not one near-miss survived — which is the half of the claim a picture makes look obvious. So: one linear condition in two unknowns does not pin them down. It thins the whole plane to a line. That line is the complete list of pairs that satisfy it — every one of them, and nothing else.

12:14Two points are enough to draw it, and the upright case is a line too. A second condition draws a second line, and answering both at once means being on both lists. Which is the crossing point — here, four rides and two games. But crossing is one of three things a pair of lines can do, and which one it does is the real question the pair is asking.

12:40The geometry is not a picture laid over the algebra. It is the same object, seen from the side.