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Middle Years

2.01 Adding and subtracting 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.

$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 pronumerals and the same logic applies.


Definition: Like terms
Two algebraic terms are called like terms if they have exactly the same combination of variables.
This includes the exponents: $x$x and $x^2$x2 are not the same variables, in the same way that $4$4 and $4^2$42 are not equal.


Breaking it down

Let's look at the expression $9x+4y-5x+2y$9x+4y5x+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
  • 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$9x5x=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+4y5x+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.


Definition: Collect 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:


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$12n9m7n.

Question 3

Simplify the expression $-6vw-4v^2w+2v^2w-8wv$6vw4v2w+2v2w8wv.

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