Three strings of symbols:
All three contain letters. Two have an equals sign. One has an equals sign and is true no matter what you plug in. These are three different kinds of mathematical object — the difference between a noun, a question, and a theorem. Students who blur the three end up writing things like "solve 3x + 5" or "prove x + 2 = 5" — sentences that sound reasonable until you notice they are asking the wrong verb of the wrong object. This article pins down the three categories.
Expression: a name for a number
An expression is a string of symbols — numbers, variables, operations — that names a number once you decide what each variable stands for. It has no equals sign. It is not a claim about anything. It is a recipe: plug in values for the letters and the recipe hands you a number.
- 3x + 5 is an expression. At x = 0 it names 5. At x = 2 it names 11. At x = -4 it names -7.
- a^2 + b^2 is an expression. At a = 3, b = 4 it names 25.
- \dfrac{1}{x - 1} is an expression. At x = 3 it names \tfrac{1}{2}; at x = 1 it names nothing (division by zero).
The verbs you apply to an expression are evaluate (plug in, compute) and simplify (rewrite into a shorter recipe naming the same number for every input). What you cannot do is "solve" it. There is no claim, no question — nothing to be true or false. Expressions are the nouns of algebra. Like the phrase "the temperature in Mumbai at noon," an expression names a specific number that in general depends on other values.
Equation: a claim that might be true
An equation is a statement \text{LHS} = \text{RHS}, where \text{LHS} and \text{RHS} are both expressions and the equals sign asserts that the two expressions name the same number. Whether that assertion is true depends on the values of the variables. An equation is not automatically true or false — it has a truth value at each choice of inputs.
- x + 2 = 5 is true at x = 3, false at x = 0, false at x = 100.
- x^2 = 9 is true at x = 3 and at x = -3, false everywhere else.
- x^2 = -4 is false at every real x.
The verb you apply is solve: find the set of variable values for which the claim is true. That set is the solution set. For x + 2 = 5 it is \{3\}; for x^2 = 9 it is \{-3, 3\}; for x^2 = -4 it is the empty set \varnothing. An equation is a sentence in the grammatical sense — it makes a claim that, like "it is raining in Delhi," might be true or false depending on the state of the world (the values of the variables). This is the same object explored in Expression vs Equation Toggle, where you can slide x and watch 3x + 5 = 11 flip from true to false.
Identity: a claim that is always true
An identity is a special kind of equation — one whose solution set is every possible value of the variables. The two expressions on either side name the same number no matter what you plug in.
The most famous example in school algebra:
Try any values. At a = 2, b = 3: left is 25, right is 4 + 12 + 9 = 25. At a = -1, b = 7: left is 36, right is 1 - 14 + 49 = 36. Pick a hundred random pairs and the two sides always agree — because the right side is literally what you get when you expand the left using the distributive law.
More identities you have met: a^2 - b^2 = (a - b)(a + b), \;x \cdot 1 = x, \;\sin^2 \theta + \cos^2 \theta = 1, \;2(x + 3) = 2x + 6. Each has an equals sign and two expressions, but unlike x + 2 = 5 there is no "solve" to do — the sides are equal as recipes, not just at isolated values. An identity is really a theorem about the two expressions: they are the same function of the variables, dressed in different clothes. The verb is prove — show, using algebraic rules, that the two sides must agree for every input. Once proved, you can use the identity to rewrite one side as the other whenever it suits you.
The contrast in one picture
| Object | Has =? |
True when? | Verb | What you produce |
|---|---|---|---|---|
| Expression 3x + 5 | No | — | Evaluate, simplify | A number (after plugging in) |
| Equation x + 2 = 5 | Yes | For some values | Solve | The solution set, e.g. \{3\} |
| Identity (a+b)^2 = a^2 + 2ab + b^2 | Yes | For all values | Prove, then use | A justification, then a rewriting tool |
Notice the progression. An expression has no =. An equation adds = and the claim might be true or false. An identity adds = and guarantees the claim is always true. The equals sign is the same symbol in both equation and identity — the difference is not notational; it is a difference in the scope of the claim.
Worked contrasts
The same two sides, three categories. Consider 3(x + 2) and 3x + 6. As expressions on their own, each names a number — at x = 4 both name 18, at x = -1 both name 3. Nothing to prove or solve. Write them with an equals sign: 3(x + 2) = 3x + 6. Now it is an equation — and at every x you try, both sides match. Not by coincidence; the distributive law guarantees it. So this equation is actually an identity, its solution set all of \mathbb{R}.
An equation can masquerade as an identity. 2(x + 3) = 2x + 7 looks similar. At x = 0: left = 6, right = 7. Already false. Try to solve: 2x + 6 = 2x + 7 \Rightarrow 6 = 7, never true. So the equation is false for every x — its solution set is empty, the opposite extreme from an identity.
The "=" alone does not tell you which. You cannot distinguish identity from ordinary equation just by looking at the symbols. The difference is logical: true for all values, or only for some? Find out by trying values, or by manipulating both sides to see whether they collapse to the same thing.
Why this matters in exams
Three kinds of problems use the same three objects, and mixing up the category loses marks.
- "Simplify 5x + 3 - 2x." Input is an expression; rewrite it as 3x + 3. Writing 5x + 3 - 2x = 0 and "solving for x" is a category error — you invented an equation that was not in the question.
- "Solve 2x - 5 = 7." Input is an equation; find the solution set \{6\}. Stopping at 2x = 12 is not finished — the question asked for x.
- "Prove that (x + y)^2 - (x - y)^2 = 4xy." Input is an identity; show, using algebra, that the two sides are the same expression. Plugging in x = 1, y = 1 and getting "4 = 4" is a check of one case, not a proof.
Different verbs, different objects, different kinds of answer. Keep the three categories straight and the question almost tells you what to do.
One-line summary
An expression names a number. An equation claims two expressions name the same number, and that claim may or may not be true. An identity is an equation whose claim is true for every value of the variables. The equals sign turns a pair of expressions into a claim; whether the claim is narrow (a few solutions) or universal (an identity) is what you discover when you solve or prove.