Introduction to Algebra – Basic Definitions
What is an Equation?
An equation says that two things are equal.
It will have an equals sign “=” like this: x+ 2 = 6
That equations says: what is on the left (x + 2) is equal to what is on the right (6)
Parts of an Equation
People can talk about equations, there are names for different parts (better than saying “that thingy there”!)
Here we have an equation
Polynomial
Example of a Polynomial: 3x^{2} + x – 2
A polynomial can have constants, variables and the exponents 0,1,2,3,…
And they can be combined using addition, subtraction and multiplication, … but not division!
Monomial, Binomial, Trinomial
There are special names for polynomials with 1, 2 or 3 terms:
Like Terms
Like Terms are terms whose variables (and their exponents such as the 2 in x^{2}) are the same.
In other words, terms that are “like” each other. (Note: the coefficients can be different)
Example: 2xy^{2 }, 6xy^{2} , (1/3)xy^{2 }
Are all like terms because the variables are all xy^{2}
What is a Formula?
A formula is a special type of equation that shows the relationship between different variables.
(A variable is a symbol like X or V that stands in for a number we don’t know yet).
Example: The formula for finding the volume of a box is: V = wdh
V stands for volume, w for width, d for depth and h for height.
If w=5, d=10 and h=4, then V = 5×10×4 = 200
A formula will have more than one variable. The following are all equations, but only some are formulas:
x = 2y – 7  Formula (relating x and y) 
a^{2} + b^{2} = c^{2}  Formula (relating a, b and c) 
x/2 + 7 = 0  Not a Formula (just an equation) 
Subject of a Formula
The “subject” of a formula is the single variable that everything else is equal to.
Example: in the formula: s = ut + ½ at^{2}
“s” is the subject of the formula
Changing the Subject
One of the very powerful things that Algebra can do is to “rearrange” a formula so that another variable is the subject.
Rearrange the volume of a box formula (V = wdh) so that the width is the subject:

V = wdh 

V / d = wh 

V / dh = w 

w = V / dh 
So now if you have a box with a depth of 2m, a height of 2m and a volume of 12m^{3}, you can calculate its width:
w = V / dh
w = 12m^{3} / (2m×2m) = 12/4 = 3m
Substitution
In Algebra “Substitution” means putting numbers where the letters are:
Example 1: If x=5 then what is 10/x + 4 ?
Put “5” where “x” is:
10/5 + 4 = 2 + 4 = 6
Example 2: If x=3 and y=4, then what is x^{2} + xy ?
Put “3” where “x” is, and “4” where “y” is:
3^{2} + 3×4 = 9 + 12 = 21
Example 3: If x=3 (but you don’t know “y”), then what is x^{2} + xy ?
Put “3” where “x” is:
3^{2} + 3y = 9 + 3y
(that is as far as you can get)
As that last example showed, you may not always get a number for an answer, sometimes just a simpler formula.
Negative Numbers
When substituting negative numbers, put () around them so you get the calculations right.
Example 4: If x = 2, then what is 1x+x^{2} ?
Put “(2)” where “x” is:
1 – (2) + (2)^{2} = 1 + 2 + 4 = 7