Systems of Equations

Definitions

Linear Equation

A linear equation can be put in the form

\[a_1x_1+a_2x_2+\cdots+a_nx_n=C\]

When $x_n$ and $C$ are constants.

System of Equations

A system of equations is two or more equations which are considered together.

Solution

A solution of a system of equations is values for variables that satisfy all equations.

Solution Set

The solution set of a system of equations is the set of all possible solutions to the system of equations.

Equivalent

Systems are equivalent if the they have identical solutions sets.

Consistent and Inconsistent

A system of equations is consistent if there is at least one solution and inconsistent if there are no solutions.

Dependent and Independent

A consistent system is independent if there is exactly one solution and dependent if there are infinitely many solutions.