Doing the same thing to both sides preserves equality

0:00There is an obvious way to solve an equation, and it is worth doing properly before anything replaces it. Here it is: two n plus one equals ninety nine. Try a number. See what the left hand side comes out to. If it is wrong, try another. Start with five. Two fives is ten, plus one is eleven. Eleven, against ninety nine. Nowhere near. But it is not nothing. You now know five is far too small.

0:31So go up. Ten. Two tens plus one is twenty one. Still small. Thirty. Sixty one. Getting closer. Forty. Eighty one. Fifty. Two fifties plus one is a hundred and one. And there it is: for the first time you have gone past. So the answer is under fifty, and eighty one was under it. It is somewhere between forty and fifty. Forty nine. Two forty nines is ninety eight, plus one is ninety nine.

1:05That is it. Ninety nine on both sides. Forty nine is a solution. And this method worked. Do not let anyone tell you otherwise. But look at what you have, and at what you have not. You walked six numbers out of all the numbers there are. In one to two hundred alone, that leaves a hundred and ninety four you never looked at. And here is the uncomfortable part. A search that stops one value short of the answer finds nothing at all.

1:37Stop at forty eight and you get nothing — and it cannot tell you whether there is no answer or whether you stopped too soon. That is the first thing it never tells you: when to stop. The second is worse. You found an answer. Is it the only one? Nothing you did could answer that. You tested six numbers. The rest are still out there. Hold on to that question. It does not get settled today.

2:07Now try the method on a second equation, and watch it fail outright. Five x minus four equals seven. Try two. Five twos is ten, take four, six. Just under. Try three. Fifteen take four, eleven. Just over. So the answer is between two and three, and there is no whole number between two and three. You can keep going as long as you like. Nothing from minus five hundred to five hundred works.

2:38The search is not slow here. It never lands at all. Something other than searching is needed. And you have already seen it. You just were not told it was a method. The balance, with sacks on both pans. You took the same weight off each pan, and the beam did not move. That was not a search. Nobody tried a sack of eleven and then a sack of twelve.

3:05It was one move, applied to each side at once, and it finished. Now write the two pans as what they are. The left hand side. The right hand side. The scale was never really about weight. It was about a claim that two things are the same size. So take the scale away and keep the move. Before using it on anything unknown, test it on something with nothing unknown at all.

3:33Fifteen plus eight equals twenty three. There is no letter here. Nothing to find. It is just true. Now add ten to the left hand side. And add ten to the right hand side. The left becomes thirty three. The right becomes thirty three. Still true. That is the whole property, and notice what it did not depend on. There was no unknown anywhere in it. Why does that always work? Because of what the equals sign says.

4:07Fifteen plus eight and twenty three are not two things. They are two names for one number. So when you add ten to each, you are adding ten to the same number twice. Of course you get the same answer twice. It is not a trick about letters. It is a fact about the equals sign, and it holds for adding, subtracting, multiplying and dividing. With one bar on it, and it is worth knowing.

4:35Multiply both sides by zero and you get nought equals nought, which is true no matter what the letter was. You would have thrown the question away. Every other move can be undone. That one cannot, so it is not allowed. Here is what the property buys you, on a line with no unknown in it. Somebody has already worked out this long sum, and here is what it came to.

5:03Now you are asked for the same sum without the plus eighty eight on the end. You could go back and add it all up again. Or you could take eighty eight off both sides. On the left, the plus eighty eight and the minus eighty eight cancel, and what is left is exactly the sum you were asked for. On the right, thirteen thousand three hundred and fifty three minus eighty eight is thirteen thousand two hundred and sixty five.

5:33That is the answer, and nothing in that long sum was ever worked out. The same idea, one aisle over. Here is a long product, already worked out. You are asked for the same product without the seven. Divide both sides by seven. On the left, the seven cancels, and what is left is the product you wanted. On the right, divide the answer by seven and you get eighty two thousand nine hundred and eighty four.

6:05Somebody will say: that is not doing anything to both sides, that is just cancelling a seven. It is the same act. Cancelling a factor off one side is exactly dividing that side by it, and it only stays true because you did it to the other side too. Now one where the sign matters, and this is where people slip. Another one, already worked out, and this time there is a minus a negative on the end of it.

6:37You want the same thing without that last term. The last term is minus negative sixty seven. So the undo is not subtracting sixty seven. You add negative sixty seven to both sides. On the left that cancels the term exactly. On the right it takes sixty seven off, giving seven thousand and twenty four. Say it slowly, because the words fight you: the undo for taking away a negative is adding that negative.

7:09And you can feel it is right, because taking away a negative made the total bigger, so undoing it has to make it smaller. One more, and it is here to kill an idea before it forms. A product with fractions in it, worked out already. You want it without the eight ninths. You divide both sides by eight ninths, and dividing by a fraction is multiplying by it upside down.

7:39So multiply both sides by nine eighths. On the left, eight ninths times nine eighths is one, and the term disappears. On the right, the same multiplication gives eleven thousand seven hundred and sixty over a hundred and thirteen. Nothing about the move changed. A fraction is not a different rule, it is just a different number. Now the equation that would not come. Five x minus four equals seven.

8:11The four is subtracted, so add four. To both sides. On the left, minus four plus four is nothing, and five x is left alone. On the right, seven plus four is eleven. Five x equals eleven. Look at those two lines. They are not two steps of a calculation. They are two ways of saying the same thing about x. Now x is multiplied by five, so divide by five. To both sides.

8:42x equals eleven fifths. Two moves, and it stopped. Nobody guessed anything. Eleven fifths. Not a whole number, and that is not a mistake. Check it. Put eleven fifths back where the x was. Five times eleven fifths. The five on the outside and the five underneath cancel, and you are left with eleven. Eleven minus four is seven. And the right hand side is seven. They agree. Eleven fifths is a solution.

9:16Now be careful about what that check just did. It settled eleven fifths. Try twelve fifths and you get eight, not seven, so that one is out. But the check says nothing about all the numbers you did not try. That is the same question the guessing left standing, and this video does not settle it either. What has changed is the work. It is directed now, it stops, and every line says why it was written.