A system of equations refers to a set of two or more equations that share common variables. These systems can be classified into linear or nonlinear, depending on the nature of the equations involved. Solving a system of equations involves finding the values of the variables that satisfy all equations simultaneously.

There are several methods for solving systems of equations:

  • Graphical method: Plotting the equations on a graph and finding the intersection points.
  • Substitution method: Solving one equation for one variable and substituting it into the other equation.
  • Elimination method: Adding or subtracting equations to eliminate one variable, making it easier to solve for the others.
  • Matrix method: Using matrices to represent the system and apply matrix operations to find the solution.

Systems can have:

  1. One solution: The system is consistent and independent.
  2. No solution: The system is inconsistent, typically due to parallel lines in the case of linear equations.
  3. Infinite solutions: The system is consistent but dependent, meaning the equations represent the same geometric object.

A system of equations is said to be consistent if it has at least one solution, and inconsistent if it has no solutions.

The following table illustrates a simple example of a system of linear equations:

Equation Form
2x + 3y = 5 Linear
4x - y = 3 Linear