Systems of Equations

A system of equations is a set of multiple equations that are solved together because they all contain the same variables.

Definition

A system of equations is a collection of two or more equations with the same set of unknowns. For example, the following is a system of equations in two variables, x and y:

x + y = 10
2x - y = 3

The solution to a system of equations is the set of variable values that satisfies all the equations in the system simultaneously.

Methods of Solving

There are three common methods for solving systems of linear equations:

  1. Substitution: Solve one equation for one variable and then substitute that expression into the other equation.
  2. Elimination (or addition): Add or subtract the equations to eliminate one of the variables, making it possible to solve for the other variable.
  3. Matrix method: Express the system of equations as a matrix and then use various operations to simplify the matrix.

Applications

Systems of equations are used in many areas of mathematics and science, as well as in fields such as engineering, economics, and computer science. They are used to model and solve problems where multiple quantities are interconnected.