IGCSE Maths / Algebraic equations
Expressions, equations and identities
Edexcel International GCSE Maths (4MA1), chapter 5: Algebraic equations, specification point 2.1a-f. Free lesson.
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Expressions, equations, identities and formulae are different kinds of algebraic statements. Recognising which one you have tells you whether to simplify, solve or substitute.
How to tell expressions, equations, identities and formulae apart
An expression is a piece of algebra with no equals sign, such as 3x + 5. You simplify it. An equation has an equals sign and is true for particular values, so you solve it. An identity is true for every value, so there is nothing to find. A formula is an equation that tells you how to work out one quantity from others. Expression: 3x + 5. Simplify it. Equation: 3x + 5 = 14. Solve it. Identity: 2(x + 3) = 2x + 6. Nothing to solve. Formula: A = πr². Substitute values into it.
How to check whether a statement is an identity
An equation may be true for some values, while an identity is true for all values. You can test a statement by substituting values for the variable. If both sides agree for every value, it is an identity. In this example, the right side is the left side expanded, so they agree for any x.
How to identify an equation and its solution
A statement that works for one particular value, but not for every value, is an equation. The value that makes it true is called its solution. Checking a different value can show that the statement is not an identity.
What to do with an expression
Simplifying means tidying an expression, not turning it into an equation. Like terms can be combined, but you must not add an equals sign or a value for x unless the question gives you an equation. 3x + 5x simplifies to 8x. Writing 3x + 5x = 8x = 0 adds an equation that was not in the question. There is nothing to solve in the expression.
What an equals sign tells you to do
An equals sign does not always mean you should solve for x. First identify what kind of statement you are looking at. Only an equation is solved. An identity is true for every value, and a formula is used by substituting values. 4y − 7 has no equals sign, so it is an expression. 5(x − 2) = 5x − 10 is true for every x, so it is an identity. 2x + 1 = 9 works only when x = 4, so it is an equation.