Introduction to Algebra — Summary

Algebra is a branch of mathematics that uses letters, numbers and symbols to solve problems. Letters are used when a number is unknown or can change.

1. Variables and Constants

A variable is a symbol, usually a letter such as x, y, a or b, that can take different values. A constant has a fixed numerical value.

For example, in 5x, x is the variable and 5 is the constant. The number multiplying a variable is called its numerical coefficient, while the variable itself is called the literal coefficient.

A relationship can be expressed using a variable. For example, if every row of an auditorium has 5 more seats than the previous row, the number of seats in row x can be written as:

5x

2. Variables in Rules

Algebra can be used to create general rules for shapes.

For a rectangle with length l and breadth b:

Perimeter = 2(l + b)

For a square with side s:

Perimeter = 4s

For an equilateral triangle with side a:

Perimeter = 3a

The perimeter of any closed figure is the sum of all its sides.

3. Rules from Arithmetic

Algebra can express familiar arithmetic properties using variables.

* Commutative property of addition:
    a + b = b + a
* Commutative property of multiplication:
    a × b = b × a
* Associative property of multiplication:
    (a × b) × c = a × (b × c)
* Distributive property:
    a × (b + c) = (a × b) + (a × c)

These rules allow arithmetic relationships to be written generally rather than only for particular numbers.

4. Algebraic Expressions

An algebraic expression is formed using one or more variables, constants and mathematical operations.

Examples include:

x + 3
5x - 8
2x + 5y

An ordinary arithmetic expression contains numbers and operations, while an algebraic expression may also contain variables.

A statement can be converted into an algebraic expression. For example:

* “3 added to a number” → x + 3
* “7 multiplied by a number” → 7x

5. V

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