Isolating the unknown, step by step

0:00Somewhere along the way you were handed a rule. Take it across the equals sign, and change its sign. And it works. Two y plus seven equals twenty one becomes two y equals twenty one take seven. The seven was a plus on the left. It arrives on the right as a minus. Nobody ever says why. That silence costs something, because the same instruction used on the wrong kind of piece gives the wrong answer.

0:32So here is what the rule actually is: one line of working, with the line above it rubbed out. This video puts that line back. Start with one that has a trap in it. Eleven y, plus negative five, equals sixty one. To clear that negative five, take negative five away — from both sides, because that is the only move allowed. On the left, negative five take negative five leaves nothing. Eleven y.

1:05On the right, sixty one take negative five. Slow down here. It is not fifty six. Taking away a negative makes the number bigger. Sixty six. Eleven y equals sixty six, and eleven sixes are sixty six, so y is six. Or divide each side by eleven and get the same six. Both finishes are the same move. Now one where the letter is on both sides. Six y plus seven equals four y plus twenty one.

1:40There is nothing to isolate yet, because y is in two places. So take four y off each side. The right side loses its y altogether. Two y plus seven equals twenty one. Take seven off each side. Two y equals fourteen. Halve each side. y equals seven. Check it. Six sevens and seven is forty nine. Four sevens and twenty one is forty nine. Now watch that same solve written out twice, side by side.

2:15On the left, every operation is written under both sides, the way we just did it. On the right, only the results are written down. Same equation. Same answer. Same number of lines. The difference is the working underneath, and the right hand column simply does not show it. That is the whole of the shortcut. Take it across and change the sign is not a different method. It is this method, with one line rubbed out.

2:48Look at what leaves, and what it turns into on the way. Two y plus seven equals twenty one becomes two y equals twenty one take seven. The seven was joined to its side by adding. To get rid of something that was added, you add its opposite. Seven and negative seven make nothing. So you add negative seven to the left, and you must add negative seven to the right too.

3:16The right side was twenty one. Now it is twenty one take seven. That is where the minus comes from. It is not a sign changing because it crossed a line. It is the opposite of the thing that was added. Next line. Two y equals fourteen. The two is not added to anything. It is multiplying. So follow the rule you were given: take it across, change the sign.

3:45y equals fourteen take two. y equals twelve. Put twelve back. Twice twelve is twenty four, and twenty four is not fourteen. The undo of multiplying by two is dividing by two, not subtracting two. y equals fourteen divided by two. Seven. And doubling a number and adding two to it do agree at exactly one number, which is two. That is why this mistake survives so long. One more shape. u over fifteen equals six.

4:21Here the fifteen is dividing. The undo of dividing by fifteen is multiplying by fifteen, on both sides. u equals six times fifteen. Ninety. Check it. Ninety over fifteen is six. And if you had added fifteen instead, you would have twenty one. A fifteenth of twenty one is not six. The fifteen came up from under the line as a multiplication, and no sign changed anywhere. Put the three of them next to each other.

4:56A term is joined to its side by adding, so it leaves as its opposite. A factor is joined by multiplying, so it leaves as a division. A divisor is dividing, so it leaves as a multiplication. Only the first of the three has anything to do with a sign. The plus becomes a minus because the undo of adding is adding the opposite — and for no other reason.

5:24Nothing flips because it crossed the equals sign. If that were the reason, the factor would flip too, and we have just watched it not. Now a bigger one, and three people solving it at once. Twenty eight, bracket x plus four, plus three hundred, equals one thousand. The first takes three hundred off each side, then divides each side by twenty eight, then takes four off each side. The second notices that twenty eight, three hundred and one thousand all divide by four.

6:00So they quarter the whole equation first, reaching seven brackets plus seventy five equals two hundred and fifty, and finish the same way. The third opens the bracket before anything else, reaching twenty eight x plus four hundred and twelve, then takes four hundred and twelve off each side and divides by twenty eight. Three different orders. All three land on x equals twenty one. There is no correct order. There is only the rule that each line follows from the one above by an operation applied to both sides.

6:40Two strange ones now, and they are strange in opposite directions. Build an equation that no number at all can satisfy. Take a number and add four. Take the same number and add five. Set those equal. x plus four equals x plus five. Take x off each side and you are left with four equals five. Whatever x is, the right hand side is exactly one more than the left. Always one more.

7:11So there is nothing to find. Not a hard answer — no answer. Solving can come back empty, and that is a result, not a failure. The other strange one looks almost identical, and it is not. Five s equals three s. Take three s off each side. Two s equals nothing at all. So s is zero. That is the answer people refuse to write down. It looks like the equation collapsed.

7:42It did not. Zero is a number, and it is the only number that works here. Try any other and the two sides differ. An empty answer and an answer of zero look alike on the page and are opposites. Which brings us to the cheapest habit in the subject. Three u take seven equals two u plus three. Take two u off each side, add seven to each side, and u is ten.

8:11Now check it. On the left, thirty take seven is twenty three. On the right, twenty plus three is twenty three. They agree. That check cost one substitution, and it catches exactly the sign slips this video is about. But be clear what it settled. It settled ten. Try nine and the sides come out twenty and twenty one. A check confirms the value you put in, and says nothing about any other.

8:43Two solutions now, each with one line wrong, failing in opposite directions. Three x take ten equals thirty five. Someone writes three x equals thirty five take ten. Twenty five. But the ten was being subtracted, so it arrives on the right as a plus. Forty five, and x is fifteen. That one failed to flip when it should have. Two v take four equals six. Someone writes v take four equals six take two.

9:15The two is a factor. It leaves as a division, not as a sign change. Two v equals ten, and v is five. That one flipped when it should not have. Both were the shortcut used with its reason removed, and both were caught by one substitution. The rule is fine. Just keep the line above it where you can see it.