Properties of Equations

An EQUATION is a statement which shows that two quantities are equal.
 
Here are some examples of equations:

2 = 7 - 5
3x = 9x2 - 22 + 3
25 + 7g + 12 = 9h + 13y

Although equations may contain only constants, we are mainly interested in those that contain variables, such as the second and third equations above.
 
To SOLVE an equation is to find all the values of the variables for which the equation is TRUE.

 2x = 8 

Substituting x = 4 makes the equation true, so 4 is a solution of the equation.
 
All of the values for which an equation is true are known collectively as the equation's SOLUTION SET. A solution set is often enclosed in brackets. For example, the solution set of 2x = 8 would be written as {4}.
 
Equations which have the same solution set are considered EQUIVALENT. It is not always easy to decide whether two equations are equivalent. However, by manipulating the equation we can always determine equivalence.
 
The basic rules for manipulating equations correspond to the fundamental arithmetic operations: addition, subtraction, multiplication, and division. All of the rules stem from a single concept: Every time an arithmetic operation is performed on one side of the equation, the same operation should be performed on the other side.
 
Here are the basic rules for solving equations:
 

• The same value can be added to both sides of an equation without altering its solution set.
• The same value can be subtracted from both sides of an equation without altering its solution set.
• Both sides of the equation can be multiplied by the same value without altering its solution set.
• Both sides of the equation can be divided by the same value without altering its solution set.

We can also use these rules to determine if two equations are equivalent. To do this, use the basic rules for altering equations to change one of the equations into an equivalent equation. If an equivalent equation for one equation can be obtained that is identical to the other equation, the equations are equivalent. If, however, such an equivalent equation cannot be obtained, the equations are not equivalent.