Inconsistent and Dependent Systems
Inconsistent and Dependent Systems covers how to identify special cases when solving a system of linear equations using elimination, as taught in Yoshiwara Elementary Algebra Chapter 4. When elimination produces 0 = k (k ≠ 0), the system is inconsistent with no solutions; when it produces 0 = 0, the system is dependent with infinitely many solutions. Grade 6 algebra students learn to recognize these outcomes to avoid incorrectly concluding a system has a unique solution.
Key Concepts
Property When Using Elimination to Solve a System.
1. If combining the two equations results in an equation of the form $$0x + 0y = k \quad (k \neq 0)$$ then the system is inconsistent.
2. If combining the two equations results in an equation of the form $$0x + 0y = 0$$ then the system is dependent.
Common Questions
What is an inconsistent system of equations?
An inconsistent system has no solution. When using elimination, you get a false statement like 0 = 4, meaning the two lines are parallel and never intersect.
What is a dependent system of equations?
A dependent system has infinitely many solutions. When using elimination, you get 0 = 0, meaning both equations represent the same line.
How do you identify inconsistent vs. dependent systems?
After elimination, if you get a false statement (0 = nonzero), it is inconsistent. If you get a true statement (0 = 0), it is dependent.
Where is this covered in Yoshiwara Elementary Algebra?
Inconsistent and Dependent Systems is covered in Chapter 4: Applications of Linear Equations in Yoshiwara Elementary Algebra.
What happens graphically with inconsistent and dependent systems?
Inconsistent systems show parallel lines (never intersect); dependent systems show the same line drawn twice (they overlap completely).