Collecting like terms

Collecting like terms is what we do when we simplify an expression: we add together the terms that are the same kind of thing, and we leave the rest alone. It sounds simple, and it is, as long as we remember one rule. We're not allowed to add apples and bananas.

When do I use this?

Whenever an expression has several terms of the same kind in it. If we need to simplify something like

$$2x + 2 + 4$$

we just work out as much as we can. The two numbers $2 + 4$ can be added, so it becomes

$$2x + 6$$

and then we're done, because 2 x's and a 6 can't be added together. Simplifying is a way of making maths easier to get an overview of, and collecting like terms is the main move. If you want the bigger picture of what a term is, it's in simplifying expressions.

The procedure

Let's take the expression

$$2a + 4b + 3a$$

1. Look at what kinds of term you have. Here we have $a$'s, and we have $b$'s. Those are 2 different things.
2. Find the terms that are alike. We can see that there's both $2a$ and $3a$. They're both $a$'s, so we're allowed to add them together: $2a + 3a = 5a$.
3. Leave the odd one out. The $4b$ has nothing to be added to, so it just stays as $4b$.
4. Write the result. We can simplify the expression to $5a + 4b$.

That's all there is to it. The only thing that goes wrong is when the $a$'s and the $b$'s get mixed, and that's the apples-and-bananas rule again: 2 apples and 4 bananas are not 6 of anything.

Worked examples

Numbers and a letter. Simplify $2x + 2 + 4$.

The $2$ and the $4$ are both plain numbers, so we add them: $2 + 4 = 6$. The $2x$ is left alone.

$$2x + 2 + 4 = 2x + 6$$

We actually can't work out any more than that, because the one is letters and the other is a number.

Two different letters. Simplify $2a + 4b + 3a$.

The $2a$ and the $3a$ are both $a$'s, so together they're $5a$. The $4b$ stays.

$$2a + 4b + 3a = 5a + 4b$$

Nothing to collect. Simplify $2a + 5$.

We have 2 a's and a 5. We're not allowed to mix those, so we can't make it any smaller. $2a + 5$ is already the answer.

So: the most important thing to remember when we simplify is that letters and numbers must not be mixed. We can easily end up with the answer to a simplifying question being

$$3a + 4b + 4$$

or something like it, and that's a completely finished answer.

Common mistakes

• "$2x + 5 = 7x$." That's the classic. $2x$ is 2 x's and $5$ is a number, two different things, so they can't be added. $2x + 5$ stays as it is.
• "The answer should always be one term." No. If the expression has several different kinds of thing in it, the answer does too. $5a + 4b$ and $3a + 4b + 4$ are finished.
• "You can drop the sign when you move a term." The sign in front belongs to the term. $2x$ is not the same as $-2x$, so when you gather the $x$'s, take each one's sign with it. This matters most inside equations, see simplifying both sides of an equation.

Related

The whole idea of terms and like terms is laid out in simplifying expressions. When the terms you need to collect are hiding inside brackets, open the brackets first, see expanding a single bracket.