Solution
Pre-AlgebraA solution is a value (or set of values) for the variable(s) that makes an equation or inequality true.
Definition
A solution is the value of the variable that makes an equation true. When you substitute it back in, both sides of the equation become equal.
Example
For the equation $x + 4 = 10$, the solution is $x = 6$, because $6 + 4 = 10$. Always check by substituting the answer back in.
Key Insight
Finding a solution is like finding the right key for a lock. Only the correct value will make both sides of the equation match.
Definition
A solution to an equation is a value (or ordered set of values for multi-variable equations) that satisfies the equation when substituted for the variables. For an inequality, the solution set is often an interval or union of intervals.
Example
For the system $x + y = 5$ and $x - y = 1$, the solution is the ordered pair $(3, 2)$. Verify: $3 + 2 = 5$ (true) and $3 - 2 = 1$ (true).
Key Insight
A conditional equation has exactly one solution (or finitely many), a contradiction has none, and an identity has infinitely many. Recognizing which type you have before solving saves time.
Definition
A solution to a polynomial equation $p(x) = 0$ is a root of $p$ in the domain of discourse. Over $\mathbb{C}$, the solution set has size equal to $\deg(p)$ by the Fundamental Theorem of Algebra (with multiplicity). For systems of polynomial equations, Bezout's theorem bounds the number of isolated solutions by the product of the degrees.
Example
The equation $x^4 - 1 = 0$ has $4$ solutions in $\mathbb{C}$: $1$, $-1$, $i$, $-i$ (the fourth roots of unity). Over $\mathbb{R}$, only two solutions exist: $1$ and $-1$.
Key Insight
The gap between the number of solutions over $\mathbb{R}$ and over $\mathbb{C}$ motivates the construction of algebraically closed fields. The real-number solutions of a system correspond to the real points of an algebraic variety.