Solving Equations
Solving an equation means finding the value(s) of the variable that make the equation true, using inverse operations to isolate the variable.
Definition
Solving an equation means finding the value of the variable that makes both sides equal, by using opposite (inverse) operations to peel away whatever is attached to the variable until it stands alone; whatever you do to one side of the equation you must do to the other, just like keeping a balance scale even. For linear equations, this involves undoing operations in reverse order of the order of operations (PEMDAS) while maintaining equality on both sides, a method formally justified by the substitution property of equality: if $a = b$, then $a + c = b + c$. Solving an equation over a domain $D$ more generally means finding the solution set $\{x \in D : f(x) = g(x)\}$; for linear equations over a field, Gaussian elimination solves systems efficiently, and nonlinear equations may require numerical methods such as Newton-Raphson iteration when closed-form solutions do not exist.
Example
Solve $x - 3 = 8$: add $3$ to both sides to get $x = 11$, and check that $11 - 3 = 8$. Solve $3(x + 2) = 21$: divide both sides by $3$ to get $x + 2 = 7$, then subtract $2$ to get $x = 5$. Newton-Raphson for solving $f(x) = 0$ uses $x_{n+1} = x_n - f(x_n)/f'(x_n)$; starting at $x_0 = 2$ for $f(x) = x^2 - 2$ gives $x_1 = 1.5$, $x_2 = 1.4167$, converging to $\sqrt{2} = 1.4142\ldots$
Key Insight
Whatever you do to one side of the equation you must do to the other side, just like keeping a balance scale even. Newton-Raphson converges quadratically near a simple root, meaning the number of correct decimal digits roughly doubles with each iteration, which makes it the gold standard for fast root-finding in numerical analysis.