Systems of Equations and Inequalities

A system of equations asks: for what values of the variables are all equations simultaneously true? This is the core of optimization, economics, chemistry, and engineering — finding the one point (or set of points) that satisfies every constraint at once.

For two equations in two variables, three geometric outcomes are possible: one intersection point (one solution, consistent independent), the same line (infinitely many solutions, consistent dependent), or parallel lines (no solution, inconsistent). Algebra detects which case: if solving gives a unique answer, one solution; if you reach a true statement like 0 = 0, infinitely many; if you reach a false statement like 5 = 0, no solution.

Substitution works well when one equation is already solved for a variable. Solve y = 3x − 1 for y, substitute into the second equation. Elimination works well when both equations are in standard form Ax + By = C. Multiply to make one pair of coefficients equal, then add to cancel that variable.

For three-variable systems, the augmented matrix [A|b] organizes the coefficients. Row operations (swap rows, multiply a row by a nonzero constant, add a multiple of one row to another) transform it into row echelon form — a staircase of zeros below the diagonal — then back-substitute from the bottom equation up.

Inequality systems require graphing. Each inequality's boundary line is either solid (≤ or ≥, included) or dashed (< or >, excluded). Shade the half-plane satisfying the inequality. The feasible region is the intersection of all shaded regions. Corners of the feasible region are found by solving pairs of boundary equations. In linear programming, the optimal value of an objective function occurs at one of these corners.

Nonlinear systems (where one equation is a circle, parabola, or other curve) are solved primarily by substitution. Solve the linear equation for one variable, substitute into the nonlinear equation, solve the resulting single-variable equation (which may give 0, 1, or 2 solutions), and back-substitute for the paired values.

Blueprint Pillar 3 — Technology and Society: Systems of equations and inequalities are the mathematics of constraints and optimization. Every airline flight schedule, every factory production plan, every hospital staff assignment, and every investment portfolio is solved by optimizing an objective subject to constraints — which is linear programming at scale. Understanding feasible regions and corner points is the conceptual foundation for the operations research that runs modern logistics.