5.01 Simplify expressions with integer coefficients
Introduction
In the previous year, we have looked at how to represent algebraic expressions , and how to generate equivalent expressions by applying properties of operations. We are now ready to simplify algebraic expressions.
Simplify algebraic expressions
The image represents 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 = 2 p
This is a very simple case of what is known as combining like terms. If we wanted to then add another 3 boxes of p apples, that is we want to add 3 p to 2 p, we can see that we would have a total of 5 p apples.
| \displaystyle 2 p + 3 p | \displaystyle = | \displaystyle \{p+p\} + \{p+p + p\} |
| \displaystyle = | \displaystyle p + p + p + p + p | |
| \displaystyle = | \displaystyle 5 p |
But what if we wanted to now add 4 boxes, each containing n bananas to our existing boxes of apples?
| \displaystyle 2 p + 3 p + 4 n | \displaystyle = | \displaystyle \{p+p\} + \{p+p + p\} + \{n+ n + n + n\} |
| \displaystyle = | \displaystyle p + p + p + p + p + n + n + n + n | |
| \displaystyle = | \displaystyle 5 p + 4 n |
Can we simplify this addition any further?
We cannot add 5 boxes of apples and 4 boxes of 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 boxes of Bapples? Bapples don't exist.
We cannot simplify this expression any further, because p and n are not like terms. If you replace p and n with any other variables the same logic applies.
Two algebraic terms are called like terms if they have exactly the same combination of variables.
This includes exponents: x and x^{2} are not like terms, in the same way that 4 and 4^{2} are not equal.
Let's look at the expression: 9 x + 4 y - 5 x + 2 y 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:
| 9 x | + 4 y | - 5 x | + 2 y |
|---|---|---|---|
| 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 9 x and -5 x 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.
| 9 x | - 5 x | + 4 y | + 2 y |
|---|---|---|---|
| 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\text{ groups of } x" and subtract "5\text{ groups of } x", then we will be left with "4\text{ groups of } x". That is 9 x - 5 x = 4 x.
Similarly, 4 y and 2 y are like terms, so we can add them: 4 y + 2 y = 6 y.
Putting this together, we have: 9 x + 4 y - 5 x + 2 y = 4 x + 6 y
Notice that we can't simplify 4 x + 6 y any further. The variables x and y represent different unknown values, and they are not like terms.
To combine like terms means to simplify an expression by combining all like terms together through addition and/or subtraction.
Examples
Example 1
Are the following like terms: 9y and 10y?
Example 2
Are the following like terms: 10x^{2}y and 9y^{2}x?
Example 3
Simplify the expression: 4m + 8m + 9m
Example 4
Simplify the following expression: 8 x + 6 y - 2 y - 4 x
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.
To combine like terms means to simplify an expression by combining all like terms together through addition and/or subtraction.