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.
| $2p+3p$2p+3p | $=$= | $\left(p+p\right)+\left(p+p+p\right)$(p+p)+(p+p+p) |
| $=$= | $p+p+p+p+p$p+p+p+p+p | |
| $=$= | $5p$5p |
But what if we wanted to now add $4$4 boxes, each containing $q$q apples to our existing boxes of apples?
| $2p+3p+4q$2p+3p+4q | $=$= | $\left(p+p\right)+\left(p+p+p\right)+\left(q+q+q+q\right)$(p+p)+(p+p+p)+(q+q+q+q) |
| $=$= | $p+p+p+p+p+q+q+q+q$p+p+p+p+p+q+q+q+q | |
| $=$= | $5p+4q$5p+4q |
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.
The different parts of the expressions have special names:
- Coefficient: the number being multiplied by a variable, for example, $3$3 in $3x+7$3x+7
- Constant: A part of an algebraic expression that does not change because it has no variable part, for example, $7$7 in $3x+7$3x+7
- Variable: A symbol, letter, or words representing a value that can vary depending on the context, for example, $x$x in $3x+7$3x+7
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:
| $9x$9x | $+$+$4y$4y | $-$−$5x$5x | $+$+$2y$2y | |||
| $9$9 groups of $x$x | plus $4$4 groups of $y$y | minus $5$5 groups of $x$x | plus $2$2 groups of $y$y | |||
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.
| $9x$9x | $-$−$5x$5x | $+$+$4y$4y | $+$+$2y$2y | |||
| $9$9 groups of $x$x | minus $5$5 groups of $x$x | plus $4$4 groups of $y$y | plus $2$2 groups of $y$y | |||
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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.
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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.
To collect like terms means to simplify an expression by combining all like terms together through addition and/or subtraction.
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.
| $3s+5t+2s+8t$3s+5t+2s+8t | $=$= | $3s+2s+5t+8t$3s+2s+5t+8t |
| $=$= | $5s+5t+8t$5s+5t+8t | |
| $=$= | $5s+13t$5s+13t |
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.
Practice questions
Question 1
Simplify the expression $9x+4x$9x+4x.
Question 2
Simplify the expression $12n-9m-7n$12n−9m−7n.
Question 3
Simplify the expression $-6vw-4v^2w+2v^2w-8wv$−6vw−4v2w+2v2w−8wv.