System of equations

Definition

A system of equations consists of two or more linear equations with the same set of variables. Solutions to the system are the variable values that satisfy all the equations simultaneously.

5 Must Know Facts For Your Next Test

1. A system can be solved using various methods, including substitution, elimination, Gaussian elimination, and matrix inverses.
2. In Gaussian elimination, you convert the system's augmented matrix to row-echelon form to find solutions.
3. A consistent system has at least one solution; an inconsistent system has no solutions; a dependent system has infinitely many solutions.
4. Using matrix inverses to solve a system requires that the coefficient matrix be square (same number of rows and columns) and invertible (non-zero determinant).
5. Graphically, the solutions of a system correspond to the points where the equations' graphs intersect.

Review Questions

• What methods can be used to solve a system of linear equations?
• How do you determine if a system is consistent or inconsistent?
• What conditions must be met for a matrix inverse method to work?

Related terms

Gaussian Elimination:

An algorithm for solving systems of linear equations by transforming the augmented matrix into row-echelon form.

Matrix Inverse: The inverse of a square matrix A is another matrix B such that $AB = BA = I$, where I is the identity matrix.

Consistent System:

A system of equations that has at least one solution.