Systems of Equations
A Comprehensive Lesson for Pre-University Algebra
1Introduction & Motivation
In mathematics and its applications, we frequently encounter situations where multiple unknown quantities are related by more than one condition. For example:
- Chemistry: Mixing two solutions of different concentrations to achieve a target mixture.
- Economics: Modeling supply and demand, where price and quantity must satisfy two relationships simultaneously.
- Engineering: Analyzing electrical circuits where currents at each junction must satisfy multiple constraints (Kirchhoff's laws).
- Computer Science: Linear regression in machine learning involves solving overdetermined systems of equations.
Why this matters: Systems of equations appear throughout science, engineering, economics, and computer science, making them one of the most practically important topics in all of mathematics. A major application of linear algebra is solving systems of linear equations.
2Fundamental Definitions
Definition: System of Equations
A system of equations is a set of two or more equations involving the same set of unknowns (variables). A solution to the system is a set of values for the unknowns that makes every equation in the system true simultaneously.
Formally, a solution of a system of linear equations is a list $(s_1, s_2, \ldots, s_n)$ of numbers that makes each equation true when the values are substituted for $x_1, x_2, \ldots, x_n$ respectively.