An equation is a statement of the equality of two expressions. An equation is formed when an equals sign is placed between two expressions. An algebraic equation is an equation that includes a variable.
Equations differ from expressions in that expressions can only be simplified or evaluated for a given value, not solved. Equations on the other hand, can be solved. Solving an equation involves finding the values for which the equation is true. Solving equations by learning how to manipulate the quantities or expressions in an equation is a large part of elementary algebra, which is used on some level in almost all areas of mathematics.
Below is an example of an equation.
x + 3 = 2x + 4
The solution of the equation is x = -1. We can confirm this by plugging -1 in for x:
-1 + 3 = 2(-1) + 4
2 = -2 + 4
2 = 2
Since plugging -1 in for x makes a true statement, we know that -1 is a solution to the equation. It is possible for an equation to have no solution.
x + 2 = x + 1
The above equation has no solution because there is no value for x such that the equation can be true. For the above equation to be true, 2 = 1 would have to be true, which is clearly not the case. Determining that an equation has no solution also constitutes solving the equation.
Equations can also have more than one solution. For example:
(x + 3)(x - 5) = 0
This equation has two solutions: x = -3 and x = 5. We can determine this by first dividing each side of the equation by one of the terms, which leaves the other term. Using basic arithmetic, we can then solve for the term, after which we repeat the process for the other term:
x + 3 = 0
x = -3
x - 5 = 0
x = 5
An equation that is true for all values of a variable is known as an identity. There are numerous identities in mathematics. Trigonometry is one branch of mathematics where identities are commonly used.
The above are some of the more basic aspects of equations. There are many different types or classifications of equations. Some examples include quadratic equations, polynomial equations, linear equations, parametric equations, differential equations, and many more. In the context of this section of the site, we will mostly be working with various types of algebraic equations where the main purpose is to solve the equation.