Collecting Like Terms

In Chapter five you met terms, the chunks that an expression is built from. Now you need to sort them. Two terms are called like terms when they carry exactly the same letter. Look at the green row: three a, five a and a on its own are all like terms, because every one of them is just a number of a-tiles. The number in front is allowed to be different, and that is the part you will end up adding. In the red row nothing matches. Two a and two b use different letters, so they are different kinds of tile, and the plain number three is different again. A term with no letter is called a constant.

Once you can spot like terms, collecting them is easy. Three a plus five a. Remember what those mean: three a is a plus a plus a, and five a is five of the same tiles. Push both piles together and you have eight tiles, so the answer is eight a. In practice you never draw the tiles. You simply add the numbers in front and keep the letter exactly as it was. Subtracting works the same way, so nine y take away two y leaves seven y. The one thing that never happens is the letter doubling up. Three a plus five a is eight a, not eight a a. You added tiles, not new letters.

Most expressions mix letters and plain numbers, so sort them into piles first. Take seven x minus two x plus four. The x terms go in one pile: seven x tiles, two of them crossed out, leaving five x. The plain number four goes in a pile of its own and has no partner to join, so it just stays. The simplified expression is five x plus four, and that is the finished answer. It is not nine x, because an x tile and a unit square are different shapes and can never be counted as one pile. Check it by putting a number in. At x equals three both versions give nineteen.

Signs are where marks get lost, so here is the habit that fixes it. Every term owns the sign written in front of it. Read four m plus three minus m plus five as plus four m, plus three, minus m, plus five. Now collect. The m terms are four m and minus m, and remember that a lonely m means one m, so four take one is three, giving three m. The plain numbers are plus three and plus five, which give eight. The answer is three m plus eight. Sometimes collecting takes you below zero, and that is fine. Two x minus five x is negative three x, using the integer rules from earlier in this chapter.

Here is collecting like terms doing real work. A rectangle has length two x and width three. The perimeter is the whole way round, so it is two x plus three plus two x plus three. Collect the x terms to get four x, collect the numbers to get six, and the perimeter is four x plus six. One short expression now covers every rectangle of that shape. You can prove the simplifying was honest by testing it. At x equals one the original gives ten and four x plus six also gives ten. At x equals five both give twenty six. Simplifying shortens the writing, but it never changes the value.