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