Think of algebra as being items. Standing at a street corner you may see trucks, cars and buses. If you see 3 trucks (3t) and 2 buses (2b) you can't add these together, so: 3t + 2b remains as 3t + 2b. If you see 3 cars then 2 buses then 4 more cars, you have seen a total of 7 cars and 2 buses, or 3c + 2b + 4c = 7c + 2b.
So 'like' terms (the same letter) can be added and subtracted but not different letters.
The same with numbers and terms. Numbers can be collected together and the letters can be collected together but letters can’t be added/subtracted with numbers.
We show multiplication between numbers and pronumerals as the number first then the letter.
For example, 2 × a = 2a, 5 × d = 5d and 4 × a × d = 4ad (or 4da).
When several numbers and letters are mixed up, calculate the product of the numbers first, then the letters. For example, 5e × 2g = 10eg. Note that when the same letter is featured twice a squared term will result. E.g. 3q × 5q = 15q2. A letter times itself is that letter squared.
Division can be shown with a ‘÷’ or a vinculum. A vinculum is a horizontal line that separates the numerator (top part of a fraction) from the denominator (bottom part). So you have to know both ways that a division can be written.
Simplifying algebraic expressions means reducing the expression to its simplest form. You have to find the HCF (highest common factor) and divide both top and bottom numbers by the HCF. You don’t write ‘1’ except in the top line if there are no pronumerals in that line.
An algebraic expression involves pronumerals (which are letters), like a, b, etc. Why use letters in mathematics? A letter can be used to represent different numbers, so it is a quick way to use the same 'formula' over and over.
To write expressions in words there are several words that should be understood:
addition - increase, raise, sum of, plus, and add.
subtraction - decrease, reduce, subtract, minus and take away.
division - divide, over, find the quotient (answer after division).
multiplication - times, product of, multiply, lots of
Substitution is replacing a 'letter' with a number. A number before a letter means multiply the number and the substituted number together. So if the equation is n = 2a, and a = 3. Then, 2 × 3 = 6 not 2a =23. Similarly, when you have x^2 and x = 5 this means 5^2 = 5 × 5 = 25 not 5 × 2 = 10.
Formulae and Substitution
Substitution means to replace a pronumeral 'letter' with a number. This is just like using a formula and putting the values in. Just remember that when a number is before a letter it is as if there is a multiplication sign between them.
Eg. G = 4f, find G when f = 12 min
G = 4f
= 4 × 12
G = 48 min
Note that G is written on the bottom row and includes the units, whatever they may be, in this case min (minutes).
Factorising is the opposite to expanding. So you are showing an expression using brackets. To factorise an expression you have to take the HCF outside the brackets. The HCF is the highest common factor.
Indexed numbers have a base and an index. The index tells you how many times the base is multiplied by itself. So 5² is = 5 × 5 and 5³ = 5 × 5 × 5. When a number is shown with an index it is in ‘index form’, when it is shown with the '×' between the numbers it is said to be in 'expanded' form (that means pulled apart).