Working With Equations

Equations with one variable are typically used to be solved. Enough information is provided about an unknown value that an equation can be written and solved to determine the value (or values) that meet the description. Equations with two variables are often used to express a relationship that connects the values, and solutions are sets of ordered pairs that are communicated through a graph. In either situation, it is often helpful, if not necessary, to manipulate the equation into a familar form. To do this, it is important to be aware of the ways that we can rewrite equations (and other statements)

Equivalent expressions are expression that, when the same number is substituted for a variable, have the same value.
Equivalent statements are two (or more) statements that have the same solution set. This term is often used for equations, but applies to inequalities as well.

An often used rule about statements is: "Anything you do one one side you have to do to the other." This is not completely accurate, as it is legal to rewrite one side of an equation without doing anything to the other. In general, you can replace any expression with an equivalent expression. This is often called simplification, but there are times when an expression is written with more complexity, at least temporarily.