Four words: variable, expression, equation, and identity. They get used almost interchangeably in casual conversation, and mixed up constantly by students who are otherwise doing the math correctly. They are not interchangeable, and mixing them up is how "simplify this" and "solve this" get answered with the wrong kind of answer. This section nails down all four, cheaply, before it costs you a wrong answer on something you actually knew how to do.
A variable is a labeled box you haven't opened yet. It might hold 5. It might hold -12. It might hold something you're actively trying to figure out. The letter is just the label on the box, not a hint about what's inside.
Nothing says there's only ever one box on the table, either. An expression like 3x + 2y has two labeled boxes, x and y, each holding its own independent value. Opening one tells you nothing about what's in the other. You'll meet expressions and equations with several variables constantly from here forward. The vocabulary in this section (expression, equation, identity, contradiction) applies the same way whether there's one box on the table or five.
An expression is a recipe, not a finished dish. 3x + 5 is a set of instructions: "take the box's contents, triple it, add 5." But it doesn't claim to equal anything in particular. It's just waiting for a value.
An equation is a claim you can check. 3x + 5 = 20 says "if you follow this recipe with the right box-value, you land on 20." That claim is either true (for the right x) or false (for every other x). This is the crucial difference. An expression has no truth value at all. You can't ask "is 3x+5 true?" any more than you can ask "is a sandwich true?" It's just a thing you build. An equation is a sentence, and sentences can be true or false.
An identity is a claim that's true no matter what you put in the box. 2(x+3) = 2x+6 isn't secretly hiding one special value of x that makes it work. Try x=0, x=100, x=-7: it holds every single time, because both sides are just two different ways of writing the same recipe.
There's a fourth, quieter case worth knowing about now. It looks almost identical to an ordinary equation but behaves completely differently. A contradiction is an equation that's true for no value at all, like x = x + 1. No box-value ever makes "some number" equal to "that same number, plus one." Try any value you like: it always fails. An ordinary equation has some solutions (often exactly one). An identity has all of them. A contradiction has none. Same-looking equals sign, three completely different outcomes, once you actually go looking for which x makes it true.
A variable is a symbol, typically a lowercase letter, that stands in for a number that is unknown, unspecified, or allowed to vary.
An expression is any combination of numbers, variables, and operations that represents a single value once the variables are filled in, but makes no claim of equality: 3x + 5 \qquad \qquad x^2 - 2x
An equation is a statement that two expressions are equal, joined by =: 3x + 5 = 20
An identity is an equation that is true for every value of the variable, not just some. For example, 2(x+3) = 2x + 6 holds no matter what x is.
A contradiction is an equation that is true for no value of the variable at all. For example, x = x + 1 is false for every real number substituted in. Every equation you'll encounter falls into exactly one of three outcomes once solved: finitely many solutions (the ordinary case), all real numbers (an identity), or no solutions at all (a contradiction).
An expression is a recipe waiting for x
Drag a and b — notice the graph updates instantly, the same way 3x+5 takes on a different value for every x you feed it.
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Worked example
Expression, equation, or identity?
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1
Look at three things side by side: 5x - 3, \quad 5x - 3 = 12, \quad 2x + 2x = 4x.
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2
5x - 3 has no equals sign: it's a recipe waiting for a value of x. This is an expression.
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3
5x - 3 = 12 makes a claim of equality that's only true for one specific x (namely x=3). This is an ordinary equation.
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4
2x + 2x = 4x is also an equation, but check a few values: x=0 gives 0=0 ✓, x=5 gives 20=20 ✓, x=-3 gives -12=-12 ✓. It's true for every x, because both sides are the same recipe written two ways. This is an identity.
Worked example
Spotting a contradiction
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1
Classify x + 4 = x + 9.
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There's an equals sign, so this is at least an equation, not an expression. Now check whether it has one solution, every solution, or none. Try subtracting x from both sides to see what's left.
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Subtracting x from both sides: 4 = 9. That's not a statement about x anymore at all. The variable canceled out completely, leaving a plain numeric claim.
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4
4 = 9 is simply false, and it stayed false regardless of what x was, because x disappeared from the equation entirely during the subtraction. No value of x can rescue a false numeric statement. This is a contradiction: an equation with no solution at all, matching notation you'll soon write as \varnothing for "the solution set is empty."
Solving 3x + 5 = 20
Each legal move keeps both sides equal, the same way a real balance stays level — watch the box get isolated.
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A common mix-up: "no solution" doesn't mean "wrong equation"
Reaching 4 = 9 while solving can feel like proof you messed up somewhere. After all, 4 obviously doesn't equal 9. But if every step you took was a legal move (the same thing to both sides), then a false statement at the end isn't a mistake report. It's the answer. It's telling you, correctly, that the equation you started with was asking for something impossible. No box-value was ever going to work, and the algebra just walked you to the proof of that instead of to a number. The habit to build: before assuming an error, check whether every step really was legal. If it was, "no solution" is the correct final answer, not a sign to keep searching.
Reaching a contradiction (such as 4 = 9) while solving is not evidence of an arithmetic mistake. It is a legitimate, correct outcome: it proves the original equation has no solution. The distinction matters because a student who reaches a false numeric statement often assumes they made an error and starts hunting for one that isn't there, rather than reporting the correct answer: "no solution."
Check your understanding (counts toward your progress)
While solving an equation, a student reaches the true-looking-false statement 6 = 6 after simplifying both sides. What does this indicate?
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A contradiction ends in a FALSE numeric statement (like 4=9) — 6=6 is true, the opposite situation.
Reaching a true numeric statement (with the variable having canceled out) is a legitimate, correct outcome — not evidence of an error.
The variable has vanished entirely from 6=6 — there's no "x=6" being claimed here, just a true statement with no variable left in it at all.
Why
When every step is legal and the variable cancels out leaving a TRUE numeric statement, that proves the equation holds for every value of the variable — an identity. (Compare: canceling to a FALSE numeric statement, like 4=9, proves a contradiction instead.)
Compare what a TRUE leftover numeric statement means versus what a FALSE one means once the variable cancels out.
Recall that a contradiction ends in a false statement like 4=9 — check whether 6=6 is false.
Solve 3x+4=19 for x.
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Why
Subtracting 4 from both sides gives 3x=15, then dividing by 3 gives x=5.
Correct answer
Also accepted:
Why
Isolate the x-term first by undoing the addition, then undo the multiplication.
Evaluate the expression 11x+9+15x at x=-1.
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Why
Simplifying first gives (26)x+9; at x=-1 that's (26)(-1)+9=-17.
Correct answer
Also accepted:
Why
Combine the like terms first (11x+15x=(26)x), then substitute x=-1.
Practice: Telling variable, expression, equation, and identity apart (6 problems · doesn’t affect your progress)
Which of these has NO equals sign?
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This has an equals sign — it's an equation.
Even this short statement has an equals sign — an equation.
This has an equals sign (and is in fact an identity) — not an expression.
Why
5x-3 has no equals sign — it's a recipe (expression), not a claim of equality.
5x-3
Which of these is an EXPRESSION, not an equation?
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This has an equals sign making a claim of equality — that makes it an equation, not just an expression.
Even a short statement like this is still an equation — it claims x equals 2.
This is an equation (in fact, an identity — true for every x) because it has an equals sign making a claim.
Why
An expression is a recipe with no equals sign and no claim of equality — 4x+1 just represents a value once x is filled in. The other three options all contain "=", making them equations.
4x + 1
What makes an equation an IDENTITY specifically, rather than just an ordinary equation?
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Identities can have one variable or several — the number of variables isn't what defines an identity.
Fractions have nothing to do with it — identities and ordinary equations can each involve fractions or not.
Identities CAN be "solved" — the answer is just "every real number," since every value works, rather than one specific value.
Why
An ordinary equation is only true for particular value(s) of the variable — that's what makes solving it meaningful. An identity is true universally, for every value, because both sides are really the same expression written two different ways.
Think about how many values of the variable need to work for an equation to count as an identity, versus an ordinary equation.
It's true for every possible value of the variable, not just one
Classify the statement 2x=2x+1.
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Sharing a leading term doesn't make it an identity — check what's left after subtracting 2x from both sides.
There's no value of x that works — subtracting 2x leaves 0=1, which is never true.
It has an equals sign relating two expressions — that makes it an equation, not a bare expression.
Why
Subtracting 2x from both sides leaves 0=1, a false numeric statement true for no value of x — a contradiction.
Subtract 2x from both sides and see what numeric statement is left.
A contradiction — subtracting 2x from both sides leaves 0=1, false for every x
Which of the following is TRUE about the relationship between expressions and equations?
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They're genuinely different: an expression makes no claim of equality, while an equation makes a claim you can check as true or false.
It's the other way around conceptually — an equation is built FROM two expressions joined by =, not a special case of a single expression.
This is backwards — expressions specifically have NO equals sign; that absence is what defines them as expressions rather than equations.
Why
An equation like 3x+5=20 is literally two expressions (3x+5 and 20) joined by an equals sign, making a claim that they're equal. Expressions themselves never contain an equals sign.
Think about what an equation is literally built out of on each side of the equals sign.
Every equation contains at least one expression on each side of the equals sign
A student says "every equation has exactly one solution." What's the flaw?
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Identities and contradictions are both real, common exceptions to "exactly one solution."
Plenty of equations, like 2x=10, have exactly one perfectly good solution.
Fractions have nothing to do with it — x+1=x+1 (an identity) has infinitely many solutions with no fractions in sight.
Why
An equation can have exactly one solution (the ordinary case), every real number as a solution (an identity), or no solution at all (a contradiction) — "exactly one" is only one of three possible outcomes.
Think about whether every equation in this section's vocabulary (identity, contradiction, ordinary) has exactly one solution.
Some equations (identities) have every real number as a solution, and others (contradictions) have none
Three verbs that mean three different things
The words "simplify," "evaluate," and "solve" show up constantly as instructions. Each asks for a genuinely different kind of answer.
Simplify is tidying the recipe. 2x + 3x becomes 5x, same recipe, fewer words. The box is still closed. You haven't peeked inside.
Evaluate is cooking the recipe once, for one specific box-value someone handed you. "If x = 4, what does 2x + 3 come out to?" You open the box, see a 4, and compute: 2(4)+3 = 11. One input, one number out.
Solve flips the whole question around. Instead of being handed a box-value and asked what comes out, you're handed the output and asked to figure out what box-value must have gone in. "2x + 3 = 11: what's x?" is detective work, not cooking. You work backward from the result to the cause.
Mixing these up is the single most common way students give a technically-correct- sounding answer to the wrong question. "Simplifying" when asked to solve leaves a variable still sitting in the answer. "Solving" an expression that has no equals sign is a request that doesn't even make sense, since there's no claim to check.
Simplify an expression: rewrite it in an equivalent but shorter or more standard form, with no numeric value substituted in (the variable stays a variable).
Evaluate an expression: substitute specific numbers for its variables and compute a single numeric result.
Solve an equation: find every value of the variable that makes the equation a true statement.
Worked example
The same starting line, three different instructions
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1
Given 4x - 7 + 2x, follow three different instructions on it: simplify, evaluate at x = 3, and, using the related equation 4x - 7 + 2x = 17, solve.
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2
Simplify 4x - 7 + 2x: combine the like terms 4x and 2x (both are "x" terms), leaving the constant alone: \underbrace{4x + 2x}_{\text{like terms}} - 7 = 6x - 7. Still an expression, still has x in it: that's expected, since simplifying never removes the variable.
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3
Evaluate the ORIGINAL expression at x=3: substitute and compute a single number. 4(3) - 7 + 2(3) = 12 - 7 + 6 = 11. No variable survives this one: the answer is a plain number, 11.
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4
Solve 4x - 7 + 2x = 17 (note: this instruction needs an equation, so a =17 was added): first simplify the left side the same way as step one, 6x - 7 = 17, then add 7 to both sides, 6x = 24, then divide by 6: \boxed{x = 4}. Three instructions, three completely different-shaped answers (an expression still holding x, a bare number, and a specific value of x) from what looked like nearly the same starting line.
Solving 6x − 7 = 17
This is the equation from the worked example above, after simplifying the left side — walk through the same moves here.
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Common mix-up: "simplify" doesn't mean "solve, but incomplete"
It's tempting to see a variable still sitting in an answer and assume "I'm not done yet, shouldn't this end in x = something?" But that instinct only applies to solving an equation. If the instruction was "simplify" and the starting object was a bare expression (no equals sign), then a tidied-up expression that still contains x is the finished, correct answer. There was never an equation here to isolate x from in the first place. Asking "but what does x equal?" about a simplified expression is a bit like asking "but what does the tidied recipe taste like?" Tidying the recipe and cooking it are two different actions, and only one of them was requested.
A frequent error is treating "simplify" as a weaker or partial version of "solve" — as though simplifying an expression is just an unfinished attempt at solving it. The two instructions apply to different objects (an expression has no equals sign; solving requires one) and produce different kinds of answers. Simplifying 6x - 7 correctly stops at 6x - 7; there is no equation here to solve further, and no amount of additional work turns a bare expression into a numeric or variable-isolated answer.
Where these verbs show up outside math class
This same trio (tidy, plug in, work backward) shows up constantly outside math too. A
spreadsheet formula like =A1*2+3 is an expression: it produces a value once a
number lands in cell A1, but the formula itself makes no claim about being "right" or
"wrong." Typing 10 into A1 and reading the result is evaluating. Being told "I want
the total to come out to 25, what should go in A1?" and working backward is solving.
That's the same three verbs, wearing spreadsheet clothes instead of algebra clothes.
Worked example
Evaluate forward, solve backward — same spreadsheet formula
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1
A spreadsheet cell holds the formula
=3*A1+4. First evaluate it for a given input, then solve it for a given target, and notice which direction each one runs. -
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Evaluate: someone types 5 into A1. Substitute and compute one number, the same way you'd evaluate 3x+4 at x=5: 3(5)+4 = 19. One input went in, one output came out.
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Solve: now flip the question. You want the formula's result to land on 40 — what does A1 need to hold? Write it as the equation 3x + 4 = 40 and work backward: subtract 4 from both sides, 3x = 36, then divide by 3, x = 12.
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4
Same formula, opposite direction. Evaluating started at the input (5) and produced an output (19). Solving started at a target output (40) and worked backward to the input (12) that produces it — detective work, like solving an algebraic equation, just running inside a spreadsheet cell instead of on paper.
Check your understanding (counts toward your progress)
Simplify 8x-3x. What is the coefficient on x?
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Why
8x-3x=(8-3)x=5x — coefficient 5.
Correct answer
Also accepted:
Why
Which verb describes this task: "find the value of x that makes 7x+5+14x=68 true"?
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Why
"find the value of x that makes 7x+5+14x=68 true" is exactly what "solve" means in algebra vocabulary -- the three verbs describe genuinely different tasks on the same expression.
Practice: Simplify, evaluate, or solve — reading the verb (6 problems · doesn’t affect your progress)
Simplify 5x + 2x - 3x. What is the resulting coefficient on x?
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Why
All three terms are "like terms" (each is a multiple of x): 5x + 2x - 3x = (5+2-3)x = 4x. The coefficient is 4.
Correct answer
Also accepted:
Why
"Simplify \frac{6x}{2}" should produce which kind of answer?
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That would be the answer to "evaluate," which requires a specific x-value to substitute — none was given here.
That's the answer to "solve," which requires an equation (an equals sign) — this is a bare expression.
Simplifying an expression never produces an equation out of nowhere — there was no equals sign to begin with.
Why
"Simplify" tidies an expression without substituting any value: \frac{6x}{2}=3x, still containing x.
Recall that 'simplify' never substitutes a specific value in for the variable.
A tidied expression still containing x, like 3x
"Evaluate x^2 - 2x at x = -3." What's the correct process?
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"Evaluate" means substitute the GIVEN value and compute — there's nothing to solve for here, since x's value was already handed to you.
Evaluating specifically requires substituting the given number in — skipping that step isn't evaluating at all.
Every occurrence of x needs the substitution, not just the first one — otherwise the computation doesn't reflect the actual expression.
Why
Evaluating means substituting the given value everywhere the variable appears, then computing a single number: (-3)^2 - 2(-3) = 9 - (-6) = 9+6 = 15.
Substitute -3 for every occurrence of x before doing any arithmetic, being careful with the negative sign in (-3)^2.
Substitute -3 for every x, then compute: (-3)^2 - 2(-3) = 9 + 6 = 15
Solve \frac{x}{4}=7 for x.
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Why
Multiplying both sides by 4: x=7(4)=28.
Correct answer
Also accepted:
Why
Undo the division by multiplying both sides by the same number.
"I want my spreadsheet total to come out to 25 — what should I type into A1?" is most like which verb?
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Simplifying doesn't involve a target output at all — it's about tidying a formula, not working backward from a number.
Evaluating goes forward (input to output) — this question goes the opposite direction, from a desired output back to the needed input.
This is precisely the "solve" pattern: handed a result, asked to find what input produces it.
Why
Working backward from a desired result to the input that produces it is exactly what "solve" means — whether the recipe is an algebraic expression or a spreadsheet formula.
Decide which direction the question is going — from an input to a computed output, or from a desired output back to the needed input.
Solve — working backward from a target output to the required input
A student is asked to "simplify 6x - 7" and answers "x = \frac{7}{6}." What went wrong?
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"x = \frac{7}{6}" isn't a valid response to "simplify" — there's no equation here, so there's no value of x to solve for.
The arithmetic itself isn't really the issue — the deeper problem is applying the wrong VERB. "Simplify" doesn't ask for a value of x at all.
6x-7 is already about as simplified as it can get (no like terms to combine) — it doesn't "simplify" into a value for x, since it isn't an equation.
Why
"Simplify" only applies to expressions and never isolates a variable to a single value — that's what "solve" does, and it requires an equation (an equals sign) to work with, which 6x-7 doesn't have. The correctly simplified answer is just 6x - 7 itself (already in simplest form).
Check whether the instruction 'simplify' ever asks for a value of x, or whether that's a different verb entirely.
They tried to solve, but "simplify" applies to an expression, which has no equals sign and nothing to isolate x from
Why this matters
From here on, every instruction in this course is one of these verbs, and reading the verb correctly is half of answering correctly. "Simplify \frac{6x}{2}" wants 3x. "Evaluate \frac{6x}{2} at x=5" wants 15. "Solve \frac{6x}{2} = 15" wants x = 5. Same expression, three different questions, three different kinds of answers — and now you can tell them apart on sight.
It's worth restating the whole chain once, since every later chapter leans on it. A variable is the unopened box. An expression is a recipe built out of one or more boxes, with no claim attached. An equation pins two expressions together with a claim of equality, which is either true for some values (ordinary), true for all of them (an identity), or true for none (a contradiction). And simplify, evaluate, and solve are the three different jobs you can be asked to do with all of that machinery. Get the vocabulary automatic now, on numbers this easy, and every instruction for the rest of the course reads as an unambiguous request instead of a guessing game about what kind of answer is wanted.
Where this shows up
Cooking
Scaling a recipe is ordinary algebra wearing an apron. "2 cups of flour per batch" is a variable relationship: total flour = 2b, where b is the number of batches. That's an expression, not yet a claim about any specific number of batches. Plugging in b=3 to find you need 6 cups is evaluating. Being told "I only have 5 cups of flour, how many batches can I make?" and working backward from 2b=5 to b=2.5 is solving. It's also a nice real example of an equation whose exact answer (2.5) isn't the practically useful one (you'd round down to 2 whole batches). That's a distinction that becomes important again the first time this course solves a word problem where "x = 4.5 people" needs to be reinterpreted as a whole number.
What's next
Variable, expression, equation: three words, now cleanly separated. The last section of this chapter takes every symbol met so far, from \in to =, and drills the one habit that makes it all stick: reading it out loud before ever touching a pencil.