What we discuss in this section:

• What satisfying an equation means
• Solution or root of an equation
• the variable or the unknown in an equation
• Conditional equations
• Identities
• Formulas

• An equation is a statement of equality between two expressions such as
$x^{2}=6x-9,\quad3x^{5}+4x+\sqrt{x^{3}}=12,\quad(x+1)^{2}=x^{2}+2x+1$
• If $A=B$ and $E=F$ (where $A,B,E$ and $F$ are numbers or expressions) then $A+E=B+F,$ $A\cdot E=B\cdot F,$ $\frac{A}{E}=\frac{B}{F}.$ Of course, we should exclude division by zero. In dividing an equation by an algebraic expression, we must note for what values of the letter the divisor becomes zero and exclude them from discussion.

• A number or expression that when substituted for a letter makes the equation true is said to satisfy the equation. That number or expression is called a root or a solution of the equation. To solve an equation means to find all the solutions.

• The letter with respect to which we solve an equation is called the variable or the unknown. (Also see Section 1.1)

• For example, in the equation $5x-10=0,$ $x$ is the variable or the unknown and $x=2$ is the only root or the only solution.

### Conditional Equations, Identities, and Formulas

• An equation that is true for all permissible values of the variables involved is called an identity. A permissible value is a value for which the expressions in the equation are defined.

• An equation that is true only for certain values of the variable involved is called a conditional equation or simply an equation.

• For example, the equation $x^{2}=6x-9$ is only valid when $x=3$, so it is a conditional equation, but $(x+1)^{2}=x^{2}+2x+1$ is true for all values of $x$, so it is an identity. The equation
$\frac{1}{x-2}+\frac{1}{x+1}=\frac{2x-1}{(x-2)(x+1)}$ is true for all values of $x$ excluding $x=2$ and $x=-1$. Because substituting $-1$ or 2 for $x$ leads to division by zero, these values are not permissible. So we can say this equation is true for all permissible values of $x$ and thus it is an identity.
• In identities, the equals sign $=$ is sometimes replaced by $\equiv$. For example, we may write $(x+1)^2\equiv x^2+2x+1$

• An equation that states a general fact or rule is called a formula. For example, the equation $A=\pi r^{2}$ for the area of a circle of radius $r$ is a formula.