Equation definition
An equation is a mathematical statement that says two expressions are equal. It always contains an equals sign. For example, 7 + 3 = 10 is an equation because both sides have the same value. In algebra, equations often contain an unknown value, usually written as a letter such as x, y or n. The purpose of solving an equation is to find the value of the unknown that makes the statement true.
A simple example is x + 5 = 12. This equation asks what value of x gives 12 when 5 is added. The answer is x = 7 because 7 + 5 = 12. The value x = 7 is called the solution. A solution is not just an answer written at the end; it is the value that makes the whole equation correct when substituted back into the original statement.
Equations are used in many maths topics, including algebra, graphs, functions, geometry, trigonometry, calculus, statistics and mechanics. At GCSE level, students solve linear equations, form equations from worded problems, solve simultaneous equations, rearrange formulas and work with quadratic equations. At A-Level, equations appear in functions, coordinate geometry, exponentials, logarithms, differentiation, integration, sequences and modelling.
The key idea behind solving equations is balance. Whatever you do to one side of the equation, you must do to the other side. If 3x + 4 = 19, you can subtract 4 from both sides to get 3x = 15. Then divide both sides by 3 to get x = 5. Each step keeps the equation balanced and preserves equality. This is why the equals sign should be treated as a balance point, not just a signal for the answer.
Equations can be numerical, algebraic or formula-based. A numerical equation might be 4 times 6 = 24. An algebraic equation might be 2x - 1 = 9. A formula can also be seen as an equation, such as area = length times width or speed = distance divided by time. Rearranging a formula is really solving an equation for a different letter.
Some equations have one solution, some have more than one solution, and some have no solution. For example, a linear equation usually has one solution. A quadratic equation can have two, one or no real solutions depending on the graph. An identity is different because it is true for every possible value of the variable, such as 2x + 2x = 4x. Students should learn to recognise the difference between an equation, an expression and an identity.
Checking an equation is an important exam habit. If you solve 5x - 2 = 18 and get x = 4, substitute 4 back into the original equation: 5 times 4 minus 2 equals 18, so the solution is correct. This helps students catch sign errors, arithmetic mistakes and incorrect rearrangements.
Understanding equations is important because they are the language of mathematical problem solving. Worded questions, science formulas, graphs and real-life models often become equations. Related ideas include expressions, variables, constants, coefficients, formulas, simultaneous equations, quadratic equations, inequalities, rearranging formulas and graph intersections. A strong definition helps students know what they are solving and why each step must keep both sides equal.