Equation
Pre-AlgebraAn equation is a mathematical statement that two expressions are equal, indicated by an equals sign.
Definition
An equation is a math statement that says two things are equal. It always has an equals sign (=) in the middle.
Example
$3x + 5 = 14$ is an equation. It says that $3x + 5$ and $14$ are the same value. You can solve it to find that $x = 3$.
Key Insight
Think of an equation as a balance scale. Whatever is on the left must equal whatever is on the right. Your job when solving is to keep the scale balanced while finding the unknown.
Definition
An equation is a statement asserting that two algebraic expressions have the same value. It contains an equals sign and may involve one or more variables. An equation is true for specific values of the variable(s) called solutions.
Example
$2x - 7 = 9$ is a linear equation in one variable. Adding $7$ to both sides: $2x = 16$. Dividing both sides by $2$: $x = 8$. Check: $2(8) - 7 = 9$. Correct.
Key Insight
Not all equations have exactly one solution. Some have none (contradictions like $0 = 5$), some have infinitely many (identities like $2x = 2x$), and some have exactly one (conditional equations).
Definition
An equation is a formula of the form $P = Q$, where $P$ and $Q$ are expressions over some domain. In abstract algebra, solving an equation over a ring $R$ means finding elements of $R$ satisfying the relation. Polynomial equations of degree $n$ over algebraically closed fields have exactly $n$ roots (counted with multiplicity) by the Fundamental Theorem of Algebra.
Example
The equation $x^2 - 5x + 6 = 0$ factors as $(x-2)(x-3) = 0$, giving roots $x = 2$ and $x = 3$ over $\mathbb{R}$. Over $\mathbb{C}$, every degree-$n$ polynomial equation has exactly $n$ roots.
Key Insight
The Fundamental Theorem of Algebra guarantees the completeness of $\mathbb{C}$ as a root field. This is why complex numbers were historically introduced: to ensure polynomial equations always have solutions.