Algebra

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers,algebra introduces quantities without fixed values, known as variables.This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations.

Variables

Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.

1. Variables may represent numbers whose values are not yet known. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as ${\displaystyle C=P+20}$.^[21]
2. Variables allow one to describe general problems, without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to ${\displaystyle 60\times 5=300}$ seconds. A more general (algebraic) description may state that the number of seconds, ${\displaystyle s=60\times m}$, where m is the number of minutes.
3. Variables allow one to describe mathematical relationships between quantities that may vary. For example, the relationship between the circumference, c, and diameter, d, of a circle is described by ${\displaystyle \pi =c/d}$.
4. Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as ${\displaystyle (a+b)=(b+a)}$.  

  

Equations

   An equation states that two expressions are equal using the symbol for equality,  (the equals sign). One of the most well-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:

${\displaystyle c^{2}=a^{2}+b^{2}}$

This equation states that ${\displaystyle c^{2}}$, representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by ${\displaystyle a}$ and ${\displaystyle b}$.

An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as ${\displaystyle a+b=b+a}$); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. ${\displaystyle x^{2}-1=8}$ is true only for ${\displaystyle x=3}$ and ${\displaystyle x=-3}$. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.

Another type of equation is an inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: ${\displaystyle a>b}$ where ${\displaystyle >}$ represents 'greater than', and ${\displaystyle a<b}$ where ${\displaystyle <}$ represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.