System of equations

Definition

A system of equations is a set of two or more equations with the same variables. The solutions to the system are the values of the variables that satisfy all equations simultaneously.

5 Must Know Facts For Your Next Test

1. A system can be consistent (having at least one solution) or inconsistent (having no solution).
2. Systems can be solved using various methods, such as substitution, elimination, Gaussian elimination, and matrix inverses.
3. The graphical representation of a system involves plotting each equation and finding their intersection points.
4. $$Ax = b$$ represents a linear system in matrix form, where $$A$$ is the coefficient matrix, $$x$$ is the variable vector, and $$b$$ is the constant vector.
5. For a square matrix $$A$$, if $$det(A) \neq 0$$, then $$A^{-1}$$ exists and can be used to solve the system as $$x = A^{-1}b$$.

Review Questions

• What does it mean for a system of equations to be consistent?
• How can you use Gaussian elimination to solve a system of equations?
• What is the significance of the determinant being non-zero in solving systems using inverses?

Related terms

Gaussian Elimination: A method for solving systems of linear equations by transforming the system's augmented matrix to row-echelon form using row operations.

Matrix Inverse: For a square matrix $$A$$, its inverse $$A^{-1}$$ satisfies $$AA^{-1} = I,$$ where $$I$$ is the identity matrix. It can be used to solve linear systems.

Determinant: A scalar value that can be computed from a square matrix. It indicates whether a matrix has an inverse; if it is zero, the inverse does not exist.