Crossing, parallel or lying on top: what each picture says about solutions

0:00Here are three pairs of equations. Each pair is two straight lines. The first: x minus two y equals nought, together with three x plus four y equals twenty. The second: two x plus three y equals nine, with four x plus six y equals eighteen. The third: x plus two y equals four, with two x plus four y equals twelve. They look alike. But draw them and you get three different pictures.

0:31The first pair crosses. The second turns out to be one single line, drawn twice. The third runs parallel and never meets. And the question this topic answers is not which is which. It is why there are only three pictures to choose from. Start with something that sounds like it could not possibly matter. Take two points. How many straight lines pass through both? One. Exactly one. Put your ruler on two pins and there is nothing left to decide.

1:06Now use that. Suppose two lines share a point — that happens all the time. Suppose they share a second point as well. Then both of them are lines through those two points. And there is only one such line. So they are not two lines that met twice. They are one line, written down twice. Try it. Pin two straight edges through one shared point and you can still swing them apart.

1:34Pin them through a second, and they collapse onto each other. There is nowhere else for the second one to go. That single fact settles the whole question. Count the pairs of values that satisfy both equations at once. The count could be nought — the lines never meet. It could be one — they cross somewhere. Could it be two? No. Two shared points force the lines to be the same line, and then they share every point, not two.

2:06So the moment the count reaches two it is already endless. Nought, one, or endlessly many. Not three. Not five. Not seventeen. The three pictures are not a list somebody drew up. They are the complete set of things a plane will allow. Take them one at a time, and start with the crossing pair. x minus two y equals nought, and three x plus four y equals twenty. Two lines that lean different ways. One rises gently through the origin; the other falls from left to right.

2:45Different leans means they cannot stay apart. Somewhere they must meet, and by what we just argued, once. One crossing, one pair of values — the only answer the two equations agree on. You can see the difference of lean in the coefficients themselves. Compare the x parts: one against three, so a third. Compare the y parts: minus two against four, so minus a half. A third and minus a half are different numbers — and that mismatch is the crossing, read off without drawing anything.

3:23Now the pair that never meets. x plus two y equals four, and two x plus four y equals twelve. Double the first equation, every term: two x plus four y equals eight. Set that beside the second, which says two x plus four y equals twelve. The left-hand sides are identical; the right-hand sides are eight and twelve. So a pair of values satisfying both would make one and the same quantity equal eight and equal twelve at once.

3:58Nothing does that. There is no answer at all. Parallel lines are often described as getting closer and never quite touching. They do not get closer. Measure the gap between these two: one unit. Forty units off to the left, one unit. Forty to the right, still one unit. Nothing dramatic happens far off the page. There is simply no crossing anywhere. The third case is the one that catches people out.

4:30Two x plus three y equals nine, and four x plus six y equals eighteen. Double the first, term by term. Two x doubles to four x. Three y doubles to six y. Nine doubles to eighteen. Four x plus six y equals eighteen — which is the second equation, character for character. So the second equation is the first multiplied through by two — the same statement, said louder.

5:01And multiplying an equation through by a number that is not nought never changes which pairs satisfy it. Two equations, one condition. One condition, one line. It is tempting to picture two lines lying on top of each other. Resist that. There is one line here with two names, and every point of it answers both equations. Endlessly many solutions — not because the lines were arranged carefully, but because there was only ever one line.

5:37Three counts, and each has a name. A pair with at least one solution is called consistent — so both the crossing pair and the doubled pair qualify. A pair with no solution at all is called inconsistent. That is the parallel one. And a pair whose two equations are equivalent — each one the other rewritten — is called dependent. Notice the order of that last definition. It is about the equations, not about the drawing.

6:10Dependent means one equation adds nothing the other had not already said. The coincident picture is the consequence, not the reason. And inconsistent does not mean the equations are wrong. Both can be perfectly sensible. It says only that nothing satisfies the two together. Which raises a question people get wrong more often than any other here. Is dependent the opposite of consistent? No. Dependent is a kind of consistent.

6:45Follow the definitions. A dependent pair has endlessly many solutions. Endlessly many is certainly at least one. At least one is exactly what consistent means. So every dependent pair is a consistent pair. They are not two boxes side by side; one sits inside the other. Consistent splits into two: pairs with exactly one solution, and dependent pairs with endlessly many. Inconsistent stands alone, outside both. So much for the three cases. Now do one properly, with a ruler.

7:25x plus three y equals six, and two x minus three y equals twelve. For each line you need two points. Choose convenient values of x and work out y. Rearrange the first: y is six minus x, all over three. Take x equal to nought and y comes out two. Take x equal to six and y comes out nought. Two points: nought and two, and six and nought.

7:55Rearrange the second: y is two x minus twelve, over three. Take x equal to nought and y is minus four. Take x equal to three and y is minus two. Two more points: nought and minus four, and three and minus two. Four points, two lines. Draw them. The lines cross on the horizontal axis, where x is six and y is nought. Now — and this matters — do not stop there. A crossing read off a drawing is a reading. Test it in both equations.

8:33Six plus three times nought is six. The first holds. Two times six minus three times nought is twelve. The second holds too. It satisfies both, so it is the solution: consistent, with exactly one answer. One thing to notice, because it can mislead you: the crossing here happens to be one of the four points we tabulated. That is luck. Later we draw a pair whose crossing is not a plotted point at all.

9:06Now a pair that no drawing decides. Five x minus eight y plus one equals nought, together with three x minus twenty-four fifths y plus three fifths equals nought. Fractions in the coefficients. A graph of this is careful pencil work with an uncertain answer. So do not draw it. Scale it. Multiply the second equation through by five thirds, term by term. Five thirds of three x is five x.

9:39Five thirds of minus twenty-four fifths y: the fives cancel, and minus twenty-four over three is minus eight. So minus eight y. Five thirds of three fifths: again the fives cancel, and three over three is one. Five x minus eight y plus one equals nought — the first equation exactly. One equation is the other scaled, so they are equivalent, so the pair is dependent and the solutions are endless.

10:12No graph settled that. Arithmetic did. One more, because it hides a trap. Ana is at a sale buying T-shirts and caps. She describes how many caps in two different ways. It is two fewer than twice the number of T-shirts. And it is also four fewer than four times that same number. Let x be the T-shirts and y the caps. The first description gives y equals two x minus two; the second gives y equals four x minus four.

10:48For the first line, x equal to two gives y equal to two, and x equal to nought gives minus two. For the second, x equal to nought gives minus four, and x equal to one gives nought. Draw them and they cross where x is one and y is nought — not one of those four plotted points. Check: two times one minus two is nought. Four times one minus four is nought. Both hold.

11:18So Ana bought one T-shirt and no caps at all. And here is the trap: nought caps is the answer, not the absence of one. An inconsistent pair has nothing satisfying it. This pair has exactly one, and one of its numbers happens to be nought. Now the honest caution about all of this. A drawing settled the crossing pair and Ana's purchase. What did it actually establish? It showed you where the crossing is, near enough to name; substitution proved it. The picture proposes, the algebra confirms.

11:58Because a picture shows you the span you happened to plot, and nothing outside it. Two lines whose leanings are nearly equal look parallel across any sheet of paper you own, and cross a thousand units off the edge. Which is exactly why the fractional pair was settled by scaling instead of by drawing. So use the graph to see what is going on. Do not use it as the proof.

12:27The claim at the centre of this — that a pair of lines never shares exactly two points — needs care to check. The danger is obvious once you see it. A machine that cannot count past one reports 'never two' about every pair of anything you hand it, for ever — and testing more lines will never expose it. So the counter used here was not told what a line is. It takes two conditions and a grid of points and returns every point that passes both. No cap, no stopping early.

13:04And before it went anywhere near a straight line it was put to pairs of curves built to meet at exactly two, three, four and five points. It came back with two, three, four and five. Only then were the lines handed to it: nine thousand seven hundred and thirty pairs of them. The counts it produced were nought, one, thirteen, twenty-one and twenty-five. Not one pair came back with two. None with three, four or five either.

13:36That is a measurement, not a limit of the instrument, because the instrument had already said two. Then the theorem itself. Fifty-three of those pairs shared more than one point, and every one of them turned out to be one equation scaled onto the other — one line, twice. And the other way round: five thousand four hundred and thirty-five pairs shared exactly one point, and not one of those was a scaling of the other.

14:08Which is the argument of this whole topic, checked rather than asserted. Two distinct points determine one line. That is the whole engine. It follows that two lines sharing two points share all of them, so a pair of linear equations has nought solutions, one, or endlessly many — and never exactly two. Crossing lines: one solution, and the pair is consistent. Parallel lines: no solution, and the pair is inconsistent — spotted by scaling one equation until the unknowns match and finding the constants disagree.

14:50One line written twice: endlessly many solutions, and the pair is dependent — which is a kind of consistent, not its opposite. Draw the lines when a picture helps you see. Then substitute, because that is what proves it. Three pictures, three counts. And there is no fourth.