Conditional Equations Versus Identical Equations

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 $A+E=B+F,\qquad A\cdot E=B\cdot F,\qquad \text{and}\qquad \frac{A}{E}=\frac{B}{F}.$ 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.

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.

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 $\equiv$.

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.