4.01 Adding and subtracting algebraic terms
Addition and subtraction of algebraic terms
If we have one box containing p apples, and then we get another box containing p apples:
We can write p apples plus p more apples as: \text{Number of apples}=p+p
Remember that adding the same number multiple times is the same as multiplying it. So two boxes of p apples can be written as: \text{Number of apples}=p+p=2p
This is a very simple case of what is known as collecting like terms. If we wanted to then add another 3 boxes of p apples, that is we want to add 3p to 2p, we can see that we would have a total of 5p apples.
| \displaystyle 2p+3p | \displaystyle = | \displaystyle \left(p+p\right)+\left(p+p+p\right) |
| \displaystyle = | \displaystyle p+p+p+p+p | |
| \displaystyle = | \displaystyle 5p |
But what if we wanted to now add 4 boxes, each containing q bananas to our existing boxes of apples?
| \displaystyle 2p+3p+4q | \displaystyle = | \displaystyle \left(p+p\right)+\left(p+p+p\right)+\left(q+q+q+q\right) |
| \displaystyle = | \displaystyle p+p+p+p+p+q+q+q+q | |
| \displaystyle = | \displaystyle 5p+4q |
Can we simplify this addition any further?
We can not add 5 apples and 4 bananas into one combined term, because we wouldn't have 9 boxes of apples, nor would we have 9 boxes of bananas. What would we have? 9 Bapples? Bapples don't exist.
We can not simplify this expression any further, because p and q are not like terms. Replacing p and q with any other different pronumerals and the same logic applies.
Two algebraic terms are called like terms if they have exactly the same combination of variables.
This includes the exponents: x and x^2 are not the same variables, in the same way that 4 and 4^2 are not equal.
Let's look at the expression 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 | +4y | -5x | +2y |
|---|---|---|---|
| 9\,\text{groups of}\,x | \text{plus}\,4\,\text{groups of}\,y | \text{minus}\,5\,\text{groups of}\,x | \text{plus}\,2\,\text{groups of}\,y |
Thinking about it this way, we can see that 9x and -5x are like terms (they both represent groups of the same unknown value x). We can now rearrange the equation, ensuring the sign attached the left of any term remains with it.
| 9x | -5x | +4x | +2y |
|---|---|---|---|
| 9\,\text{groups of}\,x | \text{minus}\,5\,\text{groups of}\,x | \text{plus}\,4\,\text{groups of}\,y | \text{plus}\,2\,\text{groups of}\,y |
If we have "9 groups of x" and subtract "5 groups of x", then we will be left with "4 groups of x". That is 9x-5x=4x.
Similarly, 4y and 2y are like terms, so we can add them: 4y+2y=6y.
Putting this together, we have 9x+4y-5x+2y=4x+6y.
Notice that we can't simplify 4x+6y any further. The variables x and 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.
Examples
Example 1
Simplify the expression 10x+3x.
Example 2
Simplify the expression 12n-9m-7n.
Example 3
Simplify the expression -6vw-4v^2w+2v^2w-8wv.
Two algebraic terms are called like terms if they have exactly the same combination of variables.
To collect like terms means to simplify an expression by combining all like terms together through addition and/or subtraction.