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Linear Equations and Their Solutions

Equations and Their Solutions

An expression that uses an equals sign to represent the relationship between quantities is called an equation.
In an equation, the part on the left side of the equals sign is called the left-hand side (LHS), and the part on the right side is the right-hand side (RHS). Together, they are referred to as the both sides of the equation.

For example, the statement “The perimeter of a square with a side length of

cm is

cm” can be expressed as the equation:

Consider the equation

. By substituting

, and

into the equation and checking if the LHS equals the RHS, we get:

Value of	Value of LHS	Value of RHS	True or False
			False
			False
			True
			False

An equation that is true for some values of

and false for others is called an equation in .

The variable in such an equation is referred to as the unknown.
A value of the unknown that makes the equation true is called a solution or root of the equation.
The process of finding the solution is called solving the equation.

On the other hand, an equation like

is true for all values of

. Such an equation, which is always true regardless of the value of

, is called an identity.

Properties of Equations

, the following properties hold:

Adding the same number to both sides preserves equality:

Subtracting the same number from both sides preserves equality:

Multiplying both sides by the same number preserves equality:

Dividing both sides by the same non-zero number preserves equality:

Linear Equations

Using the properties of equations, moving a term from one side of an equation to the other by changing its sign is called transposing.

For example, in the equation

, transposing the term

from the RHS to the LHS gives:

Here, the LHS

is a linear expression.

An equation that can be simplified to the form:

where there is exactly one unknown, is called a linear equation or simply a linear equation.