📖Definition
Algebra uses letters to represent numbers.
These may be in an equation or a formula.
For example x - 2 = 3 is an equation.
Here, we can work x out.
The equation tells us that if we take 2 away from x we're left with 3
So, x = 5 because 5 - 2 = 3
Method
When we multiply a letter by a number or another letter in algebra, we don't use a x sign.
So, 4 x p x q is written 4pq
And, if we multiply a letter by itself, a x a say, we write it a², which is pronounced 'a squared'
Finally, to divide using letters, we use the following method:
p ÷ 2 = p/2
An equation normally involves a letter, some numbers and an equals sign (which is where it gets its name 'equation' from).
For example: 3a - 1 = a + 5
Try to imagine the = as a seesaw or set of scales.
The two sides are equal so they balance.
If we change them, we need to keep them balanced and equal so, whatever we do to one side of the seesaw (equation), we must do to the other.
The aim is always to get just letters on one side of the = and just numbers on the other.
So, first, let's take a from both sides.
We get 3a - a - 1 = a - a + 5
So, 2a - 1 = 5
Now let's add 1 to both sides.
2a - 1 + 1 = 5 + 1
So, 2a = 6
Now let's divide both sides by 2
2a ÷ 2 = 6 ÷ 2
So, a = 3
Do you see how that worked? We eliminated the letters from one side of the = and the numbers from the other. We were then able to solve the equation.
You might also be given a formula to work out. See the example below to see how these work.
Creating a Formula or Expression
Or you might be asked to create a formula or expression for a real life situation.
For example:
Rafal has b biscuits.
Piotr has 3 more biscuits than Rafal.
Kristoff has 5 less biscuits than Rafal.
Find an expression for the total number of biscuits between the three of them.
We need to convert the words into mathematical language.
Rafal has b biscuits. That's easy.
Piotr has 3 more than Rafal ie. b + 3
Kristoff has 5 less than Rafal ie. b - 5
So we have an expression for the number of biscuits each of them has.
The total number of biscuits will simply be the sum of these three expressions ie.
b + b + 3 + b - 5 = b + b + b + 3 - 5
This simplifies to 3b - 2
✏️Example
The cost of a taxi fare in pence is calculated using the following formula:
300 + 50m where m = the distance travelled in miles.
How much does a journey of 7 miles cost in pounds?
✅Solution
Simply replace m in the formula with 7 remembering that 50m = 50 x m
So, the cost is 300 + 50 x 7.
Using BODMAS (see the topic on Mixed Calculations), work out 50 x 7 first
The cost is 300 + 350 = 650. Convert to £s for the answer.