3.01 Combining like terms

If we have one box containing $p$p apples, and then we get another box containing $p$p apples:

We can write $p$p apples plus $p$p more apples as:

Number of apples = $p+p$p+p

Remember that adding the same number multiple times is the same as multiplying it.
So two boxes of $p$p apples can be written as:

Number of apples = $p+p$p+p = $2p$2p

This is a very simple case of what is known as collecting like terms. If we wanted to then add another $3$3 boxes of $p$p apples, that is we want to add $3p$3p to $2p$2p, we can see that we would have a total of $5p$5p apples.

But what if we wanted to now add $4$4 boxes, each containing $q$q apples to our existing boxes of apples?

Can we simplify this addition any further?

We cannot collect $5$5 boxes of $p$p apples and $4$4 boxes of $q$q apples into one combined term, because we don't know how many apples are in each size of box.

We can not simplify this expression any further, because $p$p and $q$q are not like terms. Replace $p$p and $q$q with any other different variables and the same logic applies.

Breaking it down

Let's look at the expression $9x+4y-5x+2y$9x+4y−5x+2y. What does this mean, and how can we simplify it?

Remember that we leave out multiplication signs between numbers and variables. So we can read the expression as follows:

Thinking about it this way, we can see that $9x$9x and $-5x$−5x are like terms (they both represent groups of the same unknown value $x$x). We can now rearrange the equation, ensuring the sign attached the left of any term remains with it.

• If we have "$9$9 groups of $x$x" and subtract "$5$5 groups of $x$x", then we will be left with "$4$4 groups of $x$x". That is $9x-5x=4x$9x−5x=4x.

• Similarly, $4y$4y and $2y$2y are like terms, so we can add them: $4y+2y=6y$4y+2y=6y.

Putting this together, we have $9x+4y-5x+2y=4x+6y$9x+4y−5x+2y=4x+6y.

Notice that we can't simplify $4x+6y$4x+6y any further. The variables $x$x and $y$y represent different unknown values, and they are not like terms.

Worked example

Simplify the following expression:

$3s+5t+2s+8t$3s+5t+2s+8t

Think: To simplify an expression we collect all the like terms. $3s$3s and $2s$2s both have the same variable so they are like terms and we can combine them. Similarly, $5t$5t and $8t$8t are also like terms.

Do: Let's rearrange the expression and group the like terms together so we can clearly see which terms we need to sum.

Reflect: We identified like terms and then combined them until no like terms remained. We can add any of the terms together regardless of the ordering of the expression.