Conditional Equations Versus Identical Equations

An equation is a statement of equality between two expressions, such as

$ x^{2}=6x-9,\quad 3x^{5}+4x+\sqrt{x^{3}}=12,\quad (x+1)^{2}=x^{2}+2x+1. $

Not all equations behave the same way: some are true only for specific values of the variable, some are true for every permissible value, and some are never true at all. Understanding this distinction is fundamental to algebra.

Quick Reference: Types of Equations

Type	True for...	Example	Symbol
Conditional equation	Some (but not all) permissible values	$x^2 = 6x-9$ (only $x=3$)	$=$
Identity	All permissible values	$(x+1)^2 = x^2+2x+1$	$=$ or $\equiv$
Contradiction	No values	$x+1=x$	$=$ (but never satisfied)

Basic Properties of Equations

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.

Variable, Root, and Solution

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 and Identities

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 $\equiv$.

Contradictions: Equations With No Solution

A third type of equation, called a contradiction (or inconsistent equation), is one that is false for every value of the variable. For example,

$ x + 1 = x $

has no solution: subtracting $x$ from both sides gives $1=0$, which is never true. When solving an equation leads to a statement that is always false (such as $3=7$), the original equation is a contradiction and its solution set is empty.

The three types cover every possible case: a given equation is either true for no values (contradiction), for some but not all values (conditional), or for all permissible values (identity).

Formulas

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. A formula is a special kind of identity: it holds for all permissible values of the variables involved.

Frequently Asked Questions

What is the difference between a conditional equation and an identity?

A conditional equation is true only for specific values of the variable. A identity is true for every permissible value of the variable. For example, $2x=6$ is conditional (only $x=3$ satisfies it), while $2(x+1)=2x+2$ is an identity (true for all $x$).

How do you tell whether an equation is an identity or a conditional equation?

Try to solve the equation. If you arrive at a specific value (for example, $x=3$), it is conditional. If all variables cancel and you are left with a statement that is always true (such as $0=0$), the equation is an identity. If all variables cancel and you are left with a statement that is always false (such as $1=0$), it is a contradiction.

What is a contradiction in algebra?

A contradiction is an equation that has no solution. No value of the variable can make it true. Solving a contradiction always produces a false statement, such as $3=7$ or $0=5$. The solution set is empty, written $\emptyset$.

What does the symbol $\equiv$ mean in an equation?

The symbol $\equiv$ (read "is identically equal to") is used in place of $=$ to signal that the equation is an identity, that is, it holds for all permissible values of the variables. For example, writing $(x+1)^{2}\equiv x^{2}+2x+1$ emphasizes that the two sides are equal for every value of $x$.

What is the difference between an equation and a formula?

An equation is any statement of equality. A formula is an equation that expresses a general relationship or rule, such as $A=\pi r^{2}$ or $E=mc^{2}$. Formulas are identities in the sense that they hold for all permissible values of their variables, but the term "formula" emphasizes practical use rather than the distinction between conditional and identical equations.

Can an equation have more than one solution?

Yes. A conditional equation can have one solution, finitely many solutions, or infinitely many solutions that form a proper subset of all real numbers. For example, $x^{2}=4$ has two solutions ($x=2$ and $x=-2$). An identity, by contrast, is satisfied by every permissible value and is not usually described as having "many solutions" but rather as being "universally true."