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Addition and subtraction of Like Terms


In a previous lesson we have looked at the different components of an expression. In particular, we introduced the idea of like terms. We will now look at how to use the properties of like terms to simplify expressions.


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).

  • 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.



Are $5x^2$5x2 and $3x^2$3x2 like terms?


The variable part of $5x^2$5x2 is $x^2$x2.

The variable part of $3x^2$3x2 is also $x^2$x2.

So the value of each term depends on the same variable. This makes them like terms.



Can we simplify $3a+5$3a+5?


There are two terms here: $3a$3a and $5$5

The value of $3a$3a depends on the variable $a$a.

The value of $5$5 doesn't depend on any variable.  So these are not like terms and they can't be added.

$3a+5$3a+5 is as simple as it gets.


We can only combine the like terms in an expression (by adding or subtracting them). Unlike terms can't be combined and must stay separate.



Question 1

Simplify the expression $4f+9g-2g+f$4f+9g2g+f.

Think: Which terms are like terms? That is, which terms have the same variable parts?

Do: The terms $4f$4f and $f$f are like terms, and can be combined to give $4f+f=5f$4f+f=5f.

The terms $9g$9g and $-2g$2g are also like terms, and can be combined to give $9g-2g=7g$9g2g=7g.

So we have that $4f+9g-2g+f=5f+7g$4f+9g2g+f=5f+7g.


question 2

Simplify $4m+8m+9m$4m+8m+9m.


question 3

Simplify the expression $8p-5q+6-3p+5q-5$8p5q+63p+5q5.


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