Predicting which of the three you will get by comparing coefficients

0:00Here are two equations. Three x minus two y equals four, and six x minus four y equals nine. You could draw them: two points each, a ruler, and read off what happened. Do not. Look at the coefficients. Three and six. Minus two and minus four. The second equation's left-hand side is exactly twice the first's. But nine is not twice four. So these lines never meet, and the pair has no solution — and no graph was involved in finding that out.

0:36That is what this topic is: reading the picture off the numbers, before the picture exists. Before any of it, the equations have to be written the same way. Put every term on the left and nothing but nought on the right. The number in front of x we call a. The number in front of y, b. The number on its own, c. Two equations, so two of each: a-one, b-one and c-one from the first, a-two, b-two and c-two from the second.

1:10The signs come with them. If the equation reads x minus two y equals nought, then b is minus two, not two. That sounds like fussiness. Later in this video one dropped minus sign will turn a right answer into a wrong one. Now form three comparisons: a-one against a-two, b-one against b-two, c-one against c-two. Notice which numbers are paired: the x coefficient of the first with the x coefficient of the second. Not the x with the y inside one equation. Across the two, like with like.

1:52Take a crossing pair: x minus two y equals nought, with three x plus four y equals twenty. One against three. Then minus two against four, which is minus a half. A third is not minus a half — and that pair crosses. Take a pair that is one line twice: two x plus three y equals nine, with four x plus six y equals eighteen. Two against four is a half. Three against six is a half. Nine against eighteen is a half. All three agree.

2:28And a pair that never meets: x plus two y equals four, with two x plus four y equals twelve. One against two is a half. Two against four is a half. But four against twelve is a third. The first two agree and the third does not. Three patterns, three pictures. It looks like a table to memorise. It is not, and the rest of this video is why.

2:57Start with the first two comparisons agreeing: a-one over a-two equals b-one over b-two. Multiply both sides by a-two and by b-two and the divisions vanish. a-one times b-two equals a-two times b-one. That is the same statement with nothing divided — which matters more than it looks. Now rearrange each equation to y equals something. From a x plus b y equals c, subtract a x and divide by b: y equals minus a over b, times x, plus c over b.

3:38So the number that decides how steeply the line leans is minus a over b. And the two lines lean the same way exactly when minus a-one over b-one equals minus a-two over b-two. Cross-multiply that, and you get a-one b-two equals a-two b-one. The same statement again. So the first two ratios agreeing is not a rule about numbers. It is a statement about lean. And once it is about lean, the pictures follow without any more work.

4:14Two lines that lean the same way keep a fixed gap: step one unit right and both rise by the same amount, so the distance never changes. If it never changes, it never becomes nothing. They cannot meet. So the moment the first two ratios agree, a crossing is ruled out, and one thing is left undecided: whether the gap is zero. Zero gap and they are the same line, with endlessly many shared points.

4:46Any other gap and they are two lines that never meet, with none. And what settles the gap is the only thing left over — the constants. Look at what that means for the coincident pair. Two x plus three y equals nine, and four x plus six y equals eighteen. Doubling the first gives the second exactly, constant included. The second equation says nothing new. Same line. Now the parallel pair. x plus two y equals four, and two x plus four y equals twelve. Doubling the first gives two x plus four y equals eight.

5:29The left-hand sides match. Eight and twelve do not, and one quantity cannot be both at once. So there is really only one question in this whole topic, and it gets asked at two depths. Is the second equation a multiple of the first on the left only, or all the way through? All the way through and it is one line. On the left only and they never meet.

5:56And if the first two comparisons disagree? Then the leans are different, and two lines with different leans cannot stay apart. Far enough one way the steeper one is above; far enough the other way it is below. Somewhere in between they are level — and that is a crossing. Exactly one crossing, because two shared points would force them to be the same line. So a crossing needs no further check. The constants have nothing left to decide.

6:29That is why the first pattern has only one condition in it while the other two have two. Now a piece of honesty about what has just been established. The three worked pairs show something in one direction: given the picture, here is what the ratios do. Crossing lines gave unequal first ratios; coincident lines gave three equal ones; parallel lines gave two equal and a third out of step.

7:01But a prediction runs the other way: you are handed the ratios and you want the picture. Those are different statements, and a reverse does not come free. Here it does hold — and the reason is the lean argument, not the three examples. Equal first ratios means equal lean, and equal lean means no crossing. That runs from the numbers to the picture, which is the direction a prediction needs.

7:31Three worked cases would only ever have suggested it. Now the part that gets misapplied. The recipe says: form a-one over a-two, b-one over b-two, c-one over c-two. What if a-two is nought? Then there is no such ratio. You cannot divide by nothing, and the recipe as written has nothing to say. Take x plus two y equals three, together with y equals four. The second has no x in it, so a-two is nought and the first comparison does not exist.

8:09But the pair is perfectly ordinary. One line leans; the other is flat. They cross. The cross-product form handles it without blinking: a-one b-two is one times one, which is one; a-two b-one is nought times two, which is nought. One is not nought, so they cross. The same happens with the constants: if c-two is nought and c-one is not, the third ratio does not exist either — and if both are nought it reads nought over nought, which is worse.

8:45So when you meet a nought, do not force a ratio. Cross-multiply instead. Two more places the test gets misused. First: does it matter which equation you call the first? No. Swapping them turns every ratio upside down — a half becomes two, a third becomes three. But equal ratios are still equal upside down, and unequal ones still differ. The verdict cannot move. Second, and this one costs marks: the signs. Take three halves x plus five thirds y equals seven, with nine x minus ten y equals fourteen.

9:28Three halves divided by nine is a sixth. Five thirds divided by minus ten is minus a sixth. A sixth and minus a sixth are different, so the lines cross and the pair has one solution. Now rub out that minus sign. Five thirds divided by ten is a sixth, and the first two ratios match. The constants give seven over fourteen, which is a half, and a half is not a sixth — so the recipe now reports parallel lines and no solution at all.

10:05One character, and the answer inverts. The best way to be sure of a test is to run it backwards. Here is an equation: two x plus three y minus eight equals nought. Write a second that makes the pair cross. You need the leans to differ, so pick coefficients not in the ratio two to three. Three x plus two y minus eight equals nought will do. Now one that makes the lines parallel: same lean, different height. Double the left-hand side and change the constant — four x plus six y minus five equals nought.

10:46And one that makes them coincide: double the whole thing, constant and all. Four x plus six y minus sixteen equals nought. Look at what those two answers have in common. Both are rescalings of the left-hand side. The coincident one rescales the constant to match; the parallel one does not. Which is the thesis of this video, written as a construction instead of a test. Three pairs to try for yourself.

11:20First: five x minus four y plus eight equals nought, with seven x plus six y minus nine equals nought. Five over seven, against minus four over six, which is minus two thirds. Different — so the lines cross. Second: nine x plus three y plus twelve equals nought, with eighteen x plus six y plus twenty-four equals nought. A half, a half, and a half. All three agree, so it is one line written twice.

11:53Third: six x minus three y plus ten equals nought, with two x minus y plus nine equals nought. Six over two is three. Minus three over minus one is three. Those match. Ten over nine does not. First two equal, third different — so the lines never meet. Three pairs, three pictures, and not one line drawn. A word about how the checking was built, because this topic is unusually easy to confirm without testing anything.

12:29The trap is that the whole video is one recipe, and a machine that computes that recipe and then asserts it has agreed with itself and proved nothing. So four separate routes were run over the same twenty-one thousand nine hundred and forty-five pairs of equations. The cross-product criterion. The recipe as it is actually written, with its divisions. A route that rearranges both equations to y equals m x plus c and compares the leans.

13:04And a fourth that never forms a ratio at all: it takes two points of the first line, works out what the second equation gives at each of them, and interpolates. That last one hands back the crossing point, which is put into both equations — and so is a point that must NOT satisfy the second, because a check whose every point is supposed to pass cannot notice a test forced to say yes.

13:35The four routes agree on all twenty-one thousand nine hundred and forty-five: nineteen thousand nine hundred and eighty crossings, one thousand eight hundred and twenty-nine parallel, one hundred and thirty-six coincident. Except that the recipe does not agree, because on five thousand nine hundred of those pairs it cannot answer at all. A denominator was nought. That is the warning about noughts, measured rather than described. Rubbing every minus sign out changed the verdict on one thousand three hundred and thirty pairs — so it is not a rare accident.

14:16And swapping which equation comes first changed the verdict on none of them at all. Write both equations in the same form and read off a, b and c from each, signs included. Compare a with a and b with b, across the two equations. If they differ, the leans differ, and the lines cross at exactly one point. Consistent, one solution. If they agree, the leans agree and a crossing is already ruled out. Then look at the constants.

14:51In the same proportion, and the second equation is the first rescaled — one line, endlessly many solutions, dependent. Out of proportion, and the lines never meet. Inconsistent, no solution. When a coefficient is nought, cross-multiply rather than divide. One question, asked at two depths — and the three pictures fall out of it.