1.01 Operations with algebraic terms
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 pronumerals and the same logic applies.
Breaking it down
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 | |||
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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.
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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 collect like terms means to simplify an expression by combining all like terms together through addition and/or subtraction.
Worked example
example 1
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$12n−9m−7n.
Question 3
Simplify the expression $-6vw-4v^2w+2v^2w-8wv$−6vw−4v2w+2v2w−8wv.
Multiplying and dividing terms
We multiply and divide algebraic terms using this process:
- Split each term into its coefficient and its pronumerals.
- Find the product or quotient of the coefficient of the terms.
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- When multiplying, combine like factors into a power. For example, $x\times x=x^2$x×x=x2.
- When dividing, cancel any common factors. For example, $x\div x=1$x÷x=1.
- Combine the coefficient and pronumerals into one term.
Unlike adding and subtracting, when we multiply or divide algebraic terms, we can collect them into one term.
Worked examples
Example 2
Simplify $6x\times9y$6x×9y.
Think: Here we have a product of algebraic terms, so we can follow the process above to simplify this.
Do: $6x$6x has a coefficient of $6$6 and a pronumeral $x$x. $9y$9y has a coefficient $9$9 and a pronumeral $y$y.
We first want to evaluate the product of the coefficients. Here we have $6\times9=54$6×9=54.
Next we look at the pronumerals in each term. $6x$6x has $x$x but not $y$y and $9y$9y has $y$y but not $x$x. So we cannot simplify the pronumerals any further.
This leaves us with the factors $54$54, $x$x, and $y$y. We can simplify this without writing the multiplication signs to get $54xy$54xy.
Example 3
Simplify $6xz\div\left(9yz\right)$6xz÷(9yz).
Think: Here we have a quotient of algebraic terms, so we can follow the same process as above except that we divide instead of multiplying.
We can also write this division as the fraction $\frac{6xz}{9yz}$6xz9yz which will make the simplification easier.
Do: $6xz$6xz has a coefficient of $6$6 and the pronumerals $x$x and $z$z. $9yx$9yx has a coefficient $9$9 and the pronumerals $y$y and $z$z.
We first want to simplify the quotient of the coefficients. Here we have $\frac{6}{9}=\frac{2}{3}$69=23.
Next we simplify the pronumerals. If we take just the pronumeral part of the fraction above we get $\frac{xz}{yz}$xzyz. $z$z is common to both the numerator and the denominator so we can cancel out $z$z, but we can't cancel out $x$x or $y$y.
This leaves us with the factors $\frac{2}{3}$23 and $\frac{x}{y}$xy. We can simplify this into the fraction $\frac{2x}{3y}$2x3y.
We multiply and divide algebraic terms using this process:
- Split each term into its coefficient and its pronumerals.
- Find the product or quotient of the coefficient of the terms.
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- When multiplying, combine like factors into a power. For example, $x\times x=x^2$x×x=x2.
- When dividing, cancel any common factors. For example, $x\div x=1$x÷x=1.
- Combine the coefficient and pronumerals into one term.
Unlike adding and subtracting, when we multiply or divide algebraic terms, we can collect them into one term.
Practice questions
Question 4
Simplify the expression $9\times m\times n\times8$9×m×n×8.
Question 5
Simplify the expression $6u^2\times7v^8$6u2×7v8.
Question 6
Simplify the expression $\frac{63pq}{9p}$63pq9p.