System of three equations in three variables

Definition

A system of three equations in three variables consists of three linear equations, each containing three different variables. The solution is a set of values for the variables that satisfy all three equations simultaneously.

5 Must Know Facts For Your Next Test

1. The general form is $ax + by + cz = d$, $ex + fy + gz = h$, and $ix + jy + kz = l$.
2. Cramer's Rule can be used to solve these systems if the determinant of the coefficient matrix is non-zero.
3. The solution can be represented as an ordered triple $(x, y, z)$.
4. If the determinant of the coefficient matrix is zero, the system may have infinitely many solutions or no solution at all.
5. Each equation represents a plane in 3D space; their intersection point (if it exists) is the solution.

Review Questions

• What conditions must be met for a system of three equations in three variables to have a unique solution?
• How does Cramer's Rule apply to solving a system of three equations in three variables?
• What does it mean if the determinant of the coefficient matrix is zero?

Related terms

Determinant: A scalar value that can be computed from the elements of a square matrix and determines whether a system of linear equations has a unique solution.

Coefficient Matrix: A matrix formed from the coefficients of the variables in a system of linear equations.

Cramer's Rule: A theorem used to solve systems of linear equations with as many equations as unknowns, using determinants.