Equations and Identities

Updated August 2025

Understanding the distinction between equations and identities is fundamental for TMUA success. An equation holds true for specific values of a variable, whereas an identity is valid for all possible values. This section explains how to differentiate between them and how to mathematically demonstrate that algebraic expressions are equivalent through systematic simplification.

Core concept

An equation is an algebraic statement that is true only for specific values of a variable, whereas an identity is a statement that is true for all values of the variable for which the expressions are defined.

Definitions of Equations and Identities

In mathematics, it is vital to distinguish between a statement that happens to be true sometimes and a statement that is true by definition of its terms.

Equations

An equation is a statement that is true only for certain values of a variable. This means you can solve the equation to find those specific values. For example, $p^2 = 9$ is an equation because it is true only for the values $p = 3$ and $p = -3$ .

Identities

An identity is a statement that remains true for every possible value of the variable. If you attempt to solve an identity, the variables will eventually cancel out to leave a statement like $0 = 0$ , which is always true but does not help you isolate a specific value for the variable. An example of an identity is $5x + 3 = 3x + 2 + 2x + 1$ . No matter what number you substitute for $x$ , both sides will always be equal.

To explicitly indicate that a statement is an identity, mathematicians often use the symbol $\equiv$ instead of the standard $=$ sign. For instance, $3x + 2x - 4x \equiv x$ .

Showing Algebraic Equivalence

To mathematically prove that two algebraic expressions are equivalent, you must demonstrate that both sides can be reduced to the exact same form. This involves using algebraic rules like expanding brackets, collecting like terms, and finding common denominators to manipulate one side (or both) until they are identical.

Worked Example: Equation or Identity?

Question

Is the following statement an equation or an identity?

$\frac{4}{x+1} - \frac{3}{x-1} = \frac{1}{x^2-1}$

Solution

To determine the nature of the statement, we simplify the left hand side (LHS) and compare it to the right hand side (RHS). If the simplified LHS matches the RHS exactly, it is an identity. Otherwise, it is an equation.

Find a common denominator for the LHS, which is $(x+1)(x-1) = x^2 - 1$ .
Combine the fractions:

$\frac{4(x-1) - 3(x+1)}{(x+1)(x-1)}$

Expand the numerator:

$\frac{4x - 4 - 3x - 3}{x^2 - 1}$

Collect like terms:

$\frac{x - 7}{x^2 - 1}$

Now, compare the result $\frac{x - 7}{x^2 - 1}$ with the RHS $\frac{1}{x^2 - 1}$ . Because they are not identical, this is not an identity. It is an equation. We can find the value of $x$ for which it is true by setting the numerators equal:

$x - 7 = 1$ , which gives $x = 8$ .

Worked Example: Arguing for Equivalence

Question

Show that the following expressions are equivalent:

$\frac{2p}{3} + \frac{3q}{2} - \frac{5(p+q)}{6} = \frac{4q-p}{6}$

Solution

We start by simplifying the LHS by putting all terms over a common denominator of 6.

Rewrite each fraction with a denominator of 6:

$\frac{4p}{6} + \frac{9q}{6} - \frac{5(p+q)}{6}$

Combine into a single fraction:

$\frac{4p + 9q - 5(p + q)}{6}$

Expand the bracketed term in the numerator:

$\frac{4p + 9q - 5p - 5q}{6}$

Collect like terms:

$\frac{-p + 4q}{6} = \frac{4q - p}{6}$

Since the simplified LHS is exactly equal to the RHS, we have shown that the expressions are equivalent.

Key takeaways

An equation is only true for specific values of the variable and can be solved.
An identity is true for all possible values of the variable and is often denoted by the $\equiv$ symbol.
Attempting to solve an identity will result in a trivial statement like $0 = 0$ .
To prove equivalence, simplify one side of the expression step by step until it matches the other side exactly.
A statement is an equation if the simplified expressions on both sides are different.

Tips

In the TMUA, if a question asks you to find values for which an expression is 'true for all $x$ ', it is telling you the statement is an identity. You can often solve these by substituting easy values for $x$ , like $0$ or $1$ , to find unknown coefficients.

Cautions

Do not assume a statement is an identity just because it looks like a standard formula. Always simplify fully. For example, $(x+1)^2 = x^2+1$ might look like an identity to a tired student, but expanding the LHS reveals it is an equation that is only true when $x=0$ .

Insight

The Identity Theorem for polynomials states that if two polynomials of degree $n$ are equal for more than $n$ distinct values of $x$ , then they must be identical for all values of $x$ . This is a powerful way to confirm an identity without full algebraic expansion.

Frequently asked questions

What does the triple bar symbol mean in algebra?

The symbol $\equiv$ represents 'is identical to'. It indicates that the expressions on either side are equal for every possible value of the variables involved, rather than just being equal for specific solutions.

Can an identity have no solution?

No, an identity is the opposite: it is true for all values in the domain of the variable. If a statement has no solution, it is a contradiction, such as $x = x + 1$ .

How do I know which side to simplify when proving an identity?

It is generally easier to start with the more complex side and simplify it using algebraic rules like expanding brackets or combining fractions over a common denominator until it matches the simpler side.

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