Writing and simplifying algebraic expressions

What is an algebraic expression?
A symbol in algebra that is supposed to have a fixed value is called a constant, where as any other symbol in algebra that can be assigned different values are called a variable. For example 1/3, -8, pi, e etc are all constants and x, y, z, etc are all variables.
A sensible combination of constants and variables conjoined by arithmetic signs of +, -, * and / is called an algebraic expression. The parts or terms of an algebraic expn are separated by + or – sign. The constants and variables that are connected by * or / signs are deemed as one term. For example, in the algebraic expression 2x + 3xy + 5y - 7 there are four terms. 2x, 3xy, 5y and -7.

Simplifying an algebraic expression:
Simplification of an algebraic expression involved addition, subtraction, multiplication and division. So, How to Simplify an Expression? The rule of PEMDAS that we follow for simplifying arithmetic calculations is also followed in simplifying algebraic expn as well. Like terms can be added or subtracted together. Like terms are the terms that have the same set of variables with same exponents.  For example, 2x, 5x, 0.75x etc are all like terms. They can be added or subtracted and combined to one single term. Whereas, 2y, 5x, 0.75z^2, x^2 etc are all unlike terms. They cannot be added or subtracted. If an algebraic expn has brackets, then they have to be simplified first. There is no clear cut step wise process to simplify algebraic expn as there can be innumerable different types of expressions, and each can be simplified in simple ways. Let us look at some examples to understand better.

Example 1: 
Simplify: 3x + 4y – 3 + 4x + 7y + 8
Solution: 
For this type of expression it is possible to simplify by combining like terms.
Step 1: collect the like terms together
=> 3x + 4x + 4y + 7y + 8 – 3
Step 2: now combine the like terms
=> (3x+4x) + (4y +7y) + (8-3) = 7x + 11y + 5
That is your answer.

Writing an algebraic expression:
For any given situation described in words, that involves numbers, we can write the corresponding algebraic expn.
Example: 
Write an expression for “twice a number added to 1”.
Solution:
Here, suppose the number is x. Then twice the number would be 2x and that when added to one gives us 2x+1. This would be our required algebraic expn.