Algebra

New Zealand

Level 5

Lesson

Zara drives to work at a shopping centre where she earns $\$12$$12 per hour, and pays $\$2$$2 per hour for parking.

The number of hours she works each week can vary. So if we let the variable $k$`k` represent how many hours she works in a week, we get:

- Amount she earns each week = $12k$12
`k` - Amount she pays in parking each week = $2k$2
`k`

So each week, Zara earns $12k-2k$12`k`−2`k` after paying for parking fees.

Can we express $12k-2k$12`k`−2`k` more simply?

Since the value of both terms $12k$12`k` and $2k$2`k` depends on the same variable $k$`k`, they are like terms which we can combine.

We can think of $12k-2k$12`k`−2`k` as subtracting $2$2 lots of $k$`k` from $12$12 lots of $k$`k`. This leaves us with $10$10 lots of $k$`k`.

That is, $12k-2k=10k$12`k`−2`k`=10`k`

If we think about the scenario, Zara is really earning $\$10$$10 per hour that she works. For $k$`k` hours, she will earn $10k$10`k`.

In mathematics, we want to express things in simplest form. In expressions where we are adding and subtracting terms, we can look to see if there are like terms we can combine.

When looking for like terms, the key is to look to see if the variable part of the terms is the same.

$5u$5`u` and $2u$2`u` *are* like terms.

This is because $5u$5`u` is in terms of the variable $u$`u`, and $2u$2`u` is also in terms of the variable $u$`u`. So since they have the same variable (they both depend on $u$`u`), they are like terms.

$4t$4`t` and $8t^2$8`t`2 *are NOT* like terms because $4t$4`t` is in terms of variable $t$`t`, and $8t^2$8`t`2 is in terms of $t^2$`t`2.

As long as the different terms contain exactly the same variables, they are like terms, even if the order is different.

For example, $6mn$6`m``n` and $9mn$9`m``n` are like terms because the variables are the same. $m\times n$`m`×`n` is exactly the same as $n\times m$`n`×`m`.

If we were to think of a like term for $4x^2y$4`x`2`y` that has a coefficient of $-2$−2, we could say that either $-2x^2y$−2`x`2`y` or $-2yx^2$−2`y``x`2 are like terms.

Now that we know what like terms are and that we can only add and subtract like terms, we can simplify algebraic expressions. It is important to remember that the operator directly before a number determines whether that number is positive or negative. You can look over how to add and subtract with positive and negative numbers if you need to.

**Simplify **$14x+8x$14`x`+8`x`

**Think:** We can simplify this by adding the terms together because they are like terms. They both depend on $x$`x`.

**Do:** $14x+8x=22x$14`x`+8`x`=22`x`

**Question**: What algebraic expression should go in the space to make this statement true?

$15p$15`p` - ? = $5p$5`p`

**Think:** $15p$15`p` and $5p$5`p` are like terms, so how many lots of $p$`p` would we need to take away from $15p$15`p` to get back to $5p$5`p`?

**Do:** $15p-10p=5p$15`p`−10`p`=5`p`

**Question**: Simplify the expression: $10a-8b-4a-2b$10`a`−8`b`−4`a`−2`b`

**Think:** Let's group the like terms: 10a - 8b - 4a - 2b

Combining the like terms we get:

$10a-4a$10`a`−4`a` = $6a$6`a`

$-8b-2b$−8`b`−2`b` = $-10b$−10`b`

**Do:** $10a-8b-4a-2b=6a-10b$10`a`−8`b`−4`a`−2`b`=6`a`−10`b`

Are the following like terms? $4xy$4`x``y` and $10yx$10`y``x`

Yes

ANo

B

Simplify the expression:

$5m+9m+10m$5`m`+9`m`+10`m`

Simplify the expression:

$3x-\left(-4x\right)-2x$3`x`−(−4`x`)−2`x`

Generalise the properties of operations with fractional numbers and integers.