An equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.
An equation says both sides balance. Whatever you do to one side, you must do to the other, which is how you solve for the variable.
Formal definition and notation
An equation is a mathematical sentence A = B, where A and B are expressions. A number equation such as 3 + 4 = 7 is simply true or false. An equation with a variable is true for some values and false for others; a value that makes it true is a solution. Equations with the same solutions are equivalent, and the properties of equality turn an equation into simpler equivalent ones.
A formula is a special equation that gives a rule between quantities, like P = 2l + 2w.
Quick check: 3 questions
Question 1 of 3
Is 4x + 7 an equation?
Show answer and explanation
Answer: B. An equation needs an equal sign. 4x + 7 is an expression. Having a variable or numbers does not make an equation, and equations often contain unknowns.
Question 2 of 3
Which value of y makes 5y − 3 = 22 true?
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Answer: D. Add 3: 5y = 25. Divide by 5: y = 5. Check: 5(5) − 3 = 22. Choice A subtracts 3 instead of adding, and choice C stops before dividing.
Question 3 of 3
Which of these is an equation?
Show answer and explanation
Answer: C.7 = 3 + 4 states that two expressions are equal, so it is an equation (a true one). 8 + n and 6y are expressions, and 2x − 1 > 5 is an inequality.
Equation: frequently asked questions
What is the difference between an equation and an expression?
An equation has an equal sign and says two things are equal, like 2x + 3 = 11; you can solve it to find x = 4. An expression has no equal sign, like 2x + 3; you can simplify or evaluate it, but not solve it. Think of an expression as a phrase and an equation as a full sentence.
How do you solve an equation?
Get the variable alone by undoing operations in reverse order, doing the same thing to both sides each time. For 2x + 3 = 11, subtract 3 from both sides (2x = 8), then divide both sides by 2 (x = 4). Finish by substituting your answer into the original equation to check it.
Can an equation have more than one solution?
Yes. Linear equations like x + 4 = 9 have one solution, but x2 = 9 has two, 3 and −3. Some equations have none, like x + 1 = x + 2, and some are true for every number, like 2(x + 1) = 2x + 2. Equations with two variables, like y = 2x, have infinitely many solution pairs.
ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Equal SignThe equal sign (=) is the symbol that shows two expressions have the same value. In 3 + 5 = 2 + 6, both sides equal 8, so the statement is true. The equal sign does not mean “the answer comes next”; it works like a balanced scale, and a number sentence with = is called an equation.Symbols and Notation · Kindergarten, Grade 1, Grade 2FormulaA formula is a rule written with symbols that shows how quantities are related. It is an equation with two or more variables, such as A = lw for the area of a rectangle. Formulas help solve problems the same way each time: substitute the values you know, then compute the one you need.Problem Solving · Grade 5, Grade 6, Grade 7SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6InequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7, Grade 8EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1System of EquationsA system of equations is a set of two or more equations that use the same variables and are solved together. Its solution is the values that make every equation true at the same time. For two linear equations, that is the point where the two lines cross. A system can have one solution, none, or infinitely many.Algebra · Grade 8, Algebra 1Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2Inverse OperationsInverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.Operations · Grade 1, Grade 3, Grade 6Equivalent ExpressionsEquivalent expressions are expressions that have the same value for every value of the variable, even though they look different. For example, 3(x + 2) and 3x + 6 are equivalent: both equal 15 when x = 3 and 0 when x = −2. You build them with the properties of operations, such as the distributive property.Algebra · Grade 6, Grade 7, Algebra 1