Algebraic proof
In a nutshell
Algebraic proof involves using algebraic notation and manipulation to prove a mathematical statement is true. Start by finding an algebraic expression to represent different types of numbers, like even numbers, odd numbers or multiples of five. Then, use these expressions inside a proof to show whether something is true. It is also possible to disprove a statement by using a counterexample that shows a statement to be false.
Write expressions for proof
Write expressions to represent different types of numbers using algebraic notation.
Examples
Here are some examples, for any integer, $n$.
DESCRIPTION
 EXPRESSION

an even number  $2n$ 
an odd number  $2n+1$ 
a multiple of $3$  $3n$ 
an even multiple of $5$  $5(2n)=10n$ 
an odd mutliple of $7$  $7(2n+1)=14n+7$ 
Algebraic proof
Use algebraic expressions within a mathematical statement to prove something is true.
Example 1
Prove that the sum of two odd numbers is even.
Answer
First write expressions for the two odd numbers. For any integers, $m$ and $n$, write the odd numbers as
$2m+1 \space and \space 2n+1$
Perform the calculation to proof the result
$\begin{aligned}2m+1 + 2n+1 &= 2m+2n+2 \\&=2(m+n+1)\end {aligned}$
As $m$ and $n$ are integers, $m+n+1$ is also an integer. Two multiplied by any integer gives an even number.
Therefore, $\underline{2(m+n+1)}$ is an even number.
Disproof by counter example
It is possible to disprove a statement by finding an example to show that the statement is false.
Example 2
Give a counter example to disprove the statement 'the product of two prime numbers is always odd'.
Answer
Pick two prime numbers which  when multiplied  would give an even answer. As two is the only even prime number, multiply this number with any other prime number.
$2 \times 5 = 10$
$10$ is not an odd number so
The statement 'The product of two prime numbers is always odd' is false.