System of Linear Equations Concepts

Linear Equation:

-An equation that can be written in the form (ax + b = c), where (a, b,) and (c) are constants, and (x) is the variable.

System of linear equations:

-A set of two or more linear equations in the same variables.

Solution to a system of linear equations:

-A set of values for the variables that make all the equations in the system true.

Consistent system of linear equations:

-A system that has at least one solution.

Inconsistent system of linear equations:

-A system that has no solution.

Methods of solving systems of linear equations:

• Graphical method
• Substitution method
• Elimination method (Gaussian elimination)

## Determinant: -A scalar value that can be used to determine whether a system of linear equations has a unique solution, no solutions, or infinitely many solutions.

Cramer’s rule:

-A method for finding the solution to a system of linear equations that has a unique solution.

Rank of a matrix:

-The maximum number of linearly independent rows or columns in a matrix.

Eigenvalues and eigenvectors of a matrix:

-The eigenvalues of a square matrix are the roots of its characteristic equation, and the eigenvectors are the corresponding eigenvectors.

Applications of systems of linear equations:

-Systems of linear equations can be used to solve problems in various fields, such as engineering, physics, economics, and computer science.