Independent system

Definition

An independent system of linear equations is one in which the equations represent distinct lines that intersect at exactly one point. This means the system has a unique solution.

5 Must Know Facts For Your Next Test

1. An independent system has a unique solution where the lines intersect.
2. The determinant of the coefficient matrix of an independent system is non-zero.
3. Graphically, an independent system is represented by two lines that cross at one point.
4. In matrix form, solving an independent system using Gaussian elimination will yield a single solution.
5. If two equations in a system are scalar multiples of each other, the system is not independent.

Review Questions

• What does it mean for a system of linear equations to be independent?
• How can you determine if a given system is independent by looking at its graph?
• Explain why an independent system must have a non-zero determinant for its coefficient matrix.

Related terms

Dependent System: A dependent system consists of linear equations representing the same line, resulting in infinitely many solutions.

Inconsistent System: An inconsistent system consists of linear equations that represent parallel lines, yielding no solution.

Gaussian Elimination: $$\text{A method used to solve systems of linear equations by transforming the coefficient matrix to row-echelon form and then performing back substitution.}$$