System of three equations in three variables

Definition

A system of three equations in three variables consists of three linear equations with three unknowns, typically represented as x, y, and z. The solution to the system is the set of values for the variables that satisfy all three equations simultaneously.

5 Must Know Facts For Your Next Test

1. The system can be solved using methods such as substitution, elimination, and Cramer's Rule.
2. Cramer's Rule involves calculating determinants of matrices derived from the coefficients of the variables.
3. A unique solution exists if the determinant of the coefficient matrix is non-zero.
4. If the determinant is zero, the system may have either no solutions or infinitely many solutions.
5. Graphically, a solution corresponds to a point where three planes intersect.

Review Questions

• What are the possible outcomes (number and type of solutions) for a system of three equations in three variables?
• How do you use Cramer's Rule to solve a system of three equations in three variables?
• What does it mean if the determinant of the coefficient matrix is zero?

Related terms

Cramer's Rule: A method for solving systems of linear equations using determinants.

Determinant: A scalar value derived from a square matrix that provides important properties about the matrix and its corresponding linear system.

Coefficient Matrix: A matrix consisting of only the coefficients from a system of linear equations.