An equation asserts that two mathematical expressions have the same value. Solving it means finding all allowed assignments to its variables that make that assertion true. It converts a relationship into a constraint: a measurement, a geometric intersection, or an equilibrium model can leave an unknown value to determine.
The allowed values matter. The equation has solution set over the rational numbers, but the empty set over the integers. Over the real numbers, has two solutions, and . A mathematical identity, such as , holds for every permitted real . The familiar equality sign therefore does not by itself say whether one is checking a particular value, solving for an unknown, or stating an identity.
For a worked linear equation over the real numbers,The first step subtracts three from both sides; the second divides both sides by the nonzero number two. Reversing the steps recovers the original equation, so these are equivalent equations with the same solution set . Checking the value gives . The role of reversible operations and the classification by solution sets are described in College Algebra 2e, §2.2: Solving Linear Equations in One Variable.
Reversible operations change the graphs of the two sides but preserve the intersection's input value
. Details. When the two sides define real-valued functions and on a common domain of a function, solutions are exactly the input values at which their function graphs intersect. Equivalently, they are the zeros of a function . This gives a useful connection between symbolic algebra and geometry: rearranging an equation can simplify the comparison without changing the inputs where equality holds.
Not every operation is reversible on every domain of a function. Dividing by gives only after restricting to ; it loses the solution if that restriction is forgotten. Squaring both sides can instead introduce an extraneous solution, because distinct real numbers can have the same square. The source equation, its domain of a function, and the direction of each implication should travel together.
Two equations are equivalent on a specified domain of a function if they have the same solution set. Adding the same defined mathematical expression to both sides is reversible. Multiplying or dividing both sides by a mathematical expression is reversible only where that expression is nonzero and invertible in the arithmetic being used. In a field, every nonzero scalar is invertible; a general ring needs an additional check.
For example, and are equivalent equations over the real numbers, by the distributive property. Concluding or additionally uses the absence of zero divisors. The modular example in algebra shows why that extra inference can fail in a ring with zero divisors.
An extraneous solution satisfies a transformed equation but fails the original equation or its restrictions. The term describes a candidate generated during the calculation, rather than a true solution that can be kept in the final solution set.
Consider the real equationThe square root requires . Because the square root is nonnegative, any solution must also have , so . Squaring givesThe squared equation has candidates and . At , the original equality is . At , it is , which fails: this candidate is an extraneous solution. Thus the original solution set is exactly .
Squaring removes a sign distinction and creates a second intersection at , which fails the original equation
. Details. On the restricted domain of a function , both original sides are nonnegative, so squaring is reversible there and the two equations are equivalent. Retaining the sign restriction or checking every candidate in the original equation achieves the same final solution set. The general need to check candidates after raising both sides to a power is explained in College Algebra 2e, §2.6: Solving Radical Equations.
A mathematical identity is an equation that holds for every assignment in a specified domain of a function. For example, the distributive property gives for every real number . An identity can replace one mathematical expression by another without imposing a new restriction on that domain.
The domain remains essential. The equality holds for every real , but its left side is undefined at . Cancelling a factor does not retrospectively define the original mathematical expression at the excluded input. An identity is different from an identity element of an algebraic operation or an identity function, which are separate concepts despite sharing a name.
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An equation is a mathematical statement that asserts the equality of two expressions. It consists of two sides separated by an equal sign (=). Each side of the equation can contain numbers, variables (which represent unknown values), and mathematical operations such as addition, subtraction, multiplication, and division. For example, the equation \(2x + 3 = 7\) asserts that the expression \(2x + 3\) is equal to \(7\).