4.01 Algebra simplification

Like terms

Imagine this scenario: 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$12k
• Amount she pays in parking each week = $2k$2k

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

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

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

We can think of $12k-2k$12k−2k 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$12k−2k=10k

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$10k.

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.

Worked example

Example 1

Simplify the following if possible:

a) $5u+2u$5u+2u

Think: $5u$5u and $2u$2u are like terms. Since they are both in terms of the same variable $u$u to a single power. So we can simplify the expression.

Do: $5u+2u=7u$5u+2u=7u

b) $4t+8t^2$4t+8t2

Think: $4t$4t and $8t^2$8t2 are not like terms. Although both terms have the same variable the powers do not match.

Do: We cannot simplify this expression.

 

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

For example, $6mn$6mn and $9mn$9mn 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$4x2y that has a coefficient of $-2$−2, we could say that either $-2x^2y$−2x2y or $-2yx^2$−2yx2 are like terms.

 

example 2

Simplify the expression: $10a-8b-4a+5b$10a−8b−4a+5b

Think: Let's group the like terms: 10a - 8b - 4a + 5b and then simplify.

Do: 

$10a-8b-4a+5b$10a−8b−4a+5b	$=$=	$10a-4a-8b+5b$10a−4a−8b+5b
	$=$=	$6a-3b$6a−3b

Caution: The operator directly before a term applies to that term and needs to move with the term when rearranging. 

Practice questions

Question 1

Question 2

Multiplying and dividing terms

When we write algebraic terms such as $2y$2y, this means $2$2 groups of $y$y and there is an unwritten multiplication sign between the $2$2 and the $y$y (otherwise we would write it as $2\times y$2×y). So when we are multiplying with numbers and variables, we can multiply our numbers as normal and then multiply the variables.

Example 3

Simplify the expression $4\times r\times3\times s$4×r×3×s

Think: We don't need to write the multiplication sign between variables and we can simplify by multiplying the numbers.

Do: $4\times r\times3\times s=12rs$4×r×3×s=12rs 

example 4

What term should be written in the box to make this statement true? $8t\times\editable{}=56t$8t×=56t

Think: $8\times7=56$8×7=56. The $t$t does not have to change.

Do: $7$7 should be written in the box.

 

When we are dividing algebraic terms, we need to consider how we can simplify the numbers as well as the variables. To do this, it's often easier to write the division problem as an algebraic fraction.

For example, we could write $6x^2\div2x$6x2÷​2x as $\frac{6x^2}{2x}$6x22x​ and vice versa.

Divide the numerator and denominator by common numerical or algebraic factors to simplify the fraction.

example 5

Simplify the expression: $8x\div2$8x÷​2

Think: We could also write this as $\frac{8x}{2}$8x2​ and $8\div2=4$8÷​2=4. The $x$x does not change.

Do: $8x\div2=4x$8x÷​2=4x

Now let's look at an example where we need to divide both the numbers and the variable in the term.

example 6

Simplify the expression: $\left(-25xy\right)\div5y$(−25xy)÷​5y

Think: $-25xy\div5y=\frac{-25xy}{5y}$−25xy÷5y=−25xy5y​ and we can divide the numerator and denominator by a common factor of $5$5 and $y$y.

Do:

$-25xy\div5y$−25xy÷5y	$=$=	$\frac{-25xy}{5y}$−25xy5y​
	$=$=	$-5x$−5x

Practice questions

Question 3

Question 4

Question 5

Question 6

Expanding brackets

To expand an expression like $11\left(p+4\right)$11(p+4) or $5\left(2y-1\right)$5(2y−1) we use the distributive law:

Worked examples

Example 7

Expand and simplify the expression: $11\left(p+4\right)$11(p+4),

Think: We need to work out $11\times p$11×p and $11\times4$11×4. 

Do: 

$11(p+4)=11p+44$11(p+4)=11p+44

Example 8

Expand and simplify the expression: $5\left(2y-1\right)$5(2y−1),

Think: We need to work out $5\times2y$5×2y and $5\times\left(-1\right)$5×(−1). 

Do: 

$5(2y-1)=10y-5$5(2y−1)=10y−5

Practice questions

 Question 7

Question 8

Question 9