Collecting like terms, and what "simplest form" is for

0:00Here is a rectangle, and nobody has told you how big it is. Call the long side l and the short side b. The distance all the way round is just those four sides added. l plus b plus l plus b. That is honest, and it is also clumsy. So rearrange it. Put the two l terms together, and the two b terms. l plus l, plus b plus b.

0:28And now each pair folds into one thing. Two l, and two b. The perimeter is two l plus two b. Four terms became two, and nothing was measured, guessed or looked up. But hold on. What exactly was just claimed there? That those two expressions are equal, and that word needs pinning down. Here it means something strict. They take the same value for every replacement of the letters.

0:58Every single one. Not most, and not the ones you happened to try. So try it. Let l be three and b be four. The long form gives fourteen. The short form gives fourteen. But be careful about what that did and did not do. One matching pair proves nothing. Two l plus two b agrees with l plus three b on every square, and they are not the same expression.

1:26These two are equal because one came from the other by rules that hold for all numbers. The substitution checks the working. Now watch that same collapse happen somewhere it matters more. Pencils cost c each. Over three days you buy five, then three, then ten. So you spend five c, plus three c, plus ten c. And that is eighteen c. Which looks like adding five, three and ten, and in a sense it is.

1:58But notice that the letter was never added to anything. Five c plus three c is the distributive property read backwards. Bracket, five plus three, close bracket, times c. The numbers in front combine. The letter just sits there, being multiplied. And if a pencil costs fifty, both forms give nine hundred, which is a check rather than a reason. Erasers cost d each, and you buy four, then six, then one.

2:31Four d plus six d plus one d, and that last one has an unwritten one in front. Eleven d. Same collapse, different letter. So the whole bill is eighteen c plus eleven d. And now a question people find genuinely uncomfortable. Can that go any further? It cannot, and here is how you know rather than how you feel. Hold c completely still, change only d, and the total changes anyway.

3:02So the total is not settled by c alone, and no expression in one letter can possibly equal it. Eighteen c plus eleven d is not unfinished work. It is finished, and saying so is kinder than leaving you hunting. There is a way to see the collapse as a picture, and it is worth having. Take a rectangle of height v. Cut it with a vertical line into a piece four wide and a piece three wide.

3:33The left piece has area four v. The right piece has area three v. Together, four v plus three v. But the whole rectangle is seven wide and v high, so its area is v times seven. Same rectangle, so the same area. That is the distributive property, drawn. And it works when you take a strip away just as well as when you add one. A rectangle twelve wide and n high, with a strip four wide cut off the end.

4:05What is left is eight wide, so eight n, and also twelve n minus four n. Subtraction collects exactly as addition does. There are two words for all this, and they are worth having precisely. Terms carrying the same letter are called like terms. Five c, c and ten c are like terms. Twelve n and minus four n are like terms. Terms carrying different letters are unlike terms. Eighteen c and eleven d.

4:39Like terms may be added into one. Unlike terms may not. And that is not a rule somebody invented in order to be strict. Like terms combine because the distributive property makes them combine, at every value. Unlike terms do not, because no arithmetic fact turns two independent letters into one. So the stopping point is decided by the mathematics, and never by how short the line looks. Now the step that goes wrong most often. A bracket with a minus in front of it.

5:16You rent x chairs together with y tables. A chair costs forty, a table seventy five. When you return them, six comes back for each chair and ten for each table. So what you actually spend is forty x plus seventy five y, minus, bracket, six x plus ten y. The minus reaches everything in that second bracket. Both terms, not just the nearest one. So group the chairs with the chairs. Forty take away six, all of it times x.

5:51And the tables with the tables. Seventy five take away ten, times y. Thirty four x plus sixty five y. You could have added that second bracket instead, provided both terms inside changed sign. Same rule, from the other end. The same move handles three expressions at once. In a quiz, p is what a right answer earns and q is what a wrong one costs. Charu scores seven p minus three q, then eight p minus four q, then six p minus two q.

6:29Collect the p terms down one column. Seven, eight and six. Twenty one p. Collect the q terms down the other. Three, four and two, all being taken off. Minus nine q. Twenty one p minus nine q, and nothing was ever added across. Krishita totals twenty three p minus seven q, and the difference between them is worth looking at. Two p plus two q. The gap is itself an expression, not a number.

7:02Which is exactly right, because how far ahead she is depends on what the marks are worth. One more shape, and then the audit. A number sitting in front of a bracket. Four, bracket, x plus y, close bracket, minus y. The four multiplies everything inside it. Four x plus four y. So the whole thing reads four x plus four y minus y. Now those two y terms are like terms, so they collect.

7:34And here is the trap. That last y has no number written in front of it. It has an unwritten one, and an unwritten one is still a one. So four take away one, times y. Three y. Four x plus three y, and that same unwritten one is about to catch somebody out again. Here is a small block. Four threes, then r and s, then r and s again, then four threes.

8:03Count it row by row. Twelve, then r plus s, then r plus s, then twelve. Count it by kind instead. Two r, two s, and eight threes. Or notice the block is symmetric, so count the top half and double it. Twelve plus r plus s, doubled. Three expressions that look nothing like each other. And every one of them simplifies to two r plus two s plus twenty four.

8:31That last route only worked because the halves match, and it is worth knowing which step leaned on that. But the point stands. Different routes, one relation, and simplifying is how you see they were the same all along. So here are eleven finished simplifications, and the job is to judge them. The finish line is stated: no brackets left standing, like terms added, plain numbers added. Eight of the eleven are simply wrong.

9:05Three a plus two b, claimed as five. Five of what? Those numbers were attached to different letters. Five minus, bracket, two minus six z, claimed as three minus six z. The minus stopped at the two. Seven p minus p, claimed to leave seven p, because the second had no number in front. It had an unwritten one. One of the eleven is already correct, and hunting for a mistake in it would itself be a mistake.

9:37Two more contain no false arithmetic and are still not finished. One leaves like terms uncollected. The other is shorter than what it came from, but a bracket survives, and shortest was never the definition. And underneath it all: in simplest form the terms never outnumber the letters by more than one, and can be fewer. The letters cap the terms; they do not fix them.