What it is
An expression is numbers, variables and operations written together, like 3x + 5. It claims nothing. It has a value once you put in a number: with x = 4, 3x + 5 is 17.
An equation has an equals sign with an expression on each side: 3x + 5 = 17. It is a statement that can be true or false depending on x. To solve it is to find the x that make it true. Here that is x = 4, because 3 · 4 + 5 = 17.
Expressions, with two things you can do:
- Evaluate one: x² − 4 with x = 3 gives 9 − 4 = 5.
- Simplify it to a shorter form with the same value: 2(x + 1) = 2x + 2, for every x.
Equations, which can have different numbers of solutions:
- x = 4 is the only solution of 3x + 5 = 17.
- x = 3 or x = −3 solves x² = 9, since both 3² and (−3)² are 9.
- No solution for 2x + 3 = 2x + 5. Subtract 2x from both sides and 3 = 5 is left, which is never true.
- Every x solves 2(x + 1) = 2x + 2. An equation that holds whatever x is, is called an identity.
Not an equation: 3x + 5 (no equals sign) and 3 + 4 = 7 (no unknown, just a true equality). 3 + 4 = 8 is an equality that is false.
Why it matters
What you may do depends on which of the two you are holding.
- With an expression you change the form but never the value. That is why equals signs stand between the steps of a simplification: each line is worth the same as the one before.
- With an equation you do the same thing to both sides, so that the solutions stay the same. See the linear equation and x on both sides.
- The rules for multiplying out brackets, such as the square of a sum, are identities: they hold whatever numbers the letters stand for. That is what makes them useful for simplifying expressions.
The equation solver solves an equation step by step, and the simple equations worksheets have exercises with answers.
Common misconceptions
- “Solve the expression 3x + 5.” An expression can be evaluated or simplified but not solved; there is nothing to solve. You need an equals sign, such as 3x + 5 = 17.
- “Simplifying an equation is solving it.” Turning 3x + 5 = 17 into 3x = 12 only rewrites it. The solution arrives when x stands alone: x = 4.
- “An equation always has one solution.” x² = 9 has two, 2x + 3 = 2x + 5 has none, and 2(x + 1) = 2x + 2 has infinitely many.
- “You can do anything to one side.” Whatever you do to one side you must do to the other, or the equation no longer holds. An expression has no other side to keep level.
- “The equals sign means ‘and the answer is’.” It means both sides are equal. Written as 2 + 3 = 5 · 4 = 20, the first equality is false, since 2 + 3 is not 20, though every step was worked correctly.
Common questions
Is a formula an expression or an equation?
An equation, since it has an equals sign: A = πr² says the area equals πr². The right side πr² is an
expression. Put in r = 3 and the expression has the value 9π.
What is the difference between an equation and an identity?
An identity holds for all values of the letters, like (x + 1)² = x² + 2x + 1. An equation in the usual
sense holds only for some. Both are written with an equals sign, so the context has to tell.