In a previous lesson we have looked at the different components of an expression, and how to construct algebraic expressions.
Recall that 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 combining 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 bananas 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 can not add $5$5 apples and $4$4 bananas into one combined term, because we wouldn't have $9$9 boxes of apples, nor would we have $9$9 boxes of bananas. What would we have? $9$9 Bapples? Bapples don't exist!
We can not simplify this expression any further, because $p$p and $q$q are not like terms. Replacing $p$p and $q$q with any other different variables and the same logic applies.
Let's look at the expression $9x+4y-5x+2y$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$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$9x−5x=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+4y−5x+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.
To combine like terms means to simplify an expression by combining all like terms together through addition and/or subtraction.
Simplify the following expression:
$3s+5t+2s+8t$3s+5t+2s+8t
Think: To simplify an expression we combine all the like terms. $3s$3s and $2s$2s both have the same variables 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.
Are the following like terms: $9y$9y and $10y$10y?
Yes
No
Yes
No
Are the following like terms: $10x^2y$10x2y and $9y^2x$9y2x?
Yes
No
Yes
No
Simplify the expression: $3c+4c+7c$3c+4c+7c
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.