Lesson

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.

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.

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+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:

$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$12n9m7n.

Question 3

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

Outcomes

9.C1.3

Compare algebraic expressions using concrete, numerical, graphical, and algebraic methods to identify those that are equivalent, and justify their choices.

9.C1.4

Simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools, in different contexts.