One-Variable Equation
Pre-AlgebraA one-variable equation contains a single unknown quantity and can be solved to find one specific value (or a finite set of values) for that variable.
Definition
A one-variable equation has just one unknown letter (variable) in it. You can solve it to find out exactly what that letter equals.
Example
$3x + 7 = 19$ has just one variable, $x$. Solving it gives $x = 4$.
Key Insight
One variable means one unknown. Once you solve it, the mystery is fully resolved.
Definition
A one-variable equation involves a single variable and yields a finite solution set when solved. Linear one-variable equations (degree 1) have exactly one solution. Quadratic one-variable equations (degree 2) have up to two solutions.
Example
Linear: $5x - 3 = 12$ gives $x = 3$. Quadratic: $x^2 = 16$ gives $x = 4$ or $x = -4$ (two solutions).
Key Insight
The number of solutions depends on the degree of the equation. A degree-$n$ polynomial equation has at most $n$ real solutions and exactly $n$ complex solutions (with multiplicity).
Definition
A one-variable polynomial equation $p(x) = 0$ of degree $n$ over $\mathbb{R}$ has at most $n$ real roots and exactly $n$ roots in $\mathbb{C}$ (Fundamental Theorem of Algebra, with multiplicity). Solving methods vary by degree: formulas exist up to degree $4$ (quadratic, cubic, quartic formulas); Galois theory proves no general algebraic formula exists for degree $5$ or higher.
Example
The general quadratic $ax^2 + bx + c = 0$ is solved by $x = (-b \pm \sqrt{b^2-4ac})/(2a)$. The discriminant $b^2 - 4ac$ determines the nature of roots: positive gives two real roots, zero gives one repeated root, negative gives two complex roots.
Key Insight
Abel-Ruffini theorem: there is no general algebraic (radical) solution for polynomial equations of degree $5$ or higher. This result, proved in the early 19th century, was a watershed moment in abstract algebra and motivated Galois theory.