INDEPENDENT EQUATIONS AND INEQUALITIES
Definition Of Independent Equations And Inequalities
Independent Equations: A system of equations with exactly one solution.
Video Examples: Solving Equations and Inequalities
Example of Independent Equations and Inequalities
The system of equations given below is independent.
x + y + 3z = 12
y + z = - 4
z = 2
Solved Example on Independent Equations and Inequalities
Ques: State whether the system is consistent and independent, consistent and dependent, or inconsistent:
Choices:
A. Inconsistent
B. Consistent and independent
C. Consistent and dependent
D. Consistent
Correct Answer: B
Solution:
Step 1: [Multiply the third equation by -1/2 then,add this equation to the first equation.]
Step 2: y = - 2 [Solve for y.]
Step 3: 
[Subtracting y = - 2 in the second equation.]
Step 4: 5x + 5z = 10 [Multiply the third equation by 5.]
Step 5: 2x - 5z = - 3
5x + 5z = 10
7x = 7 [Add.]
Step 6: x = 1 [Solve for x.]
Step 7: x + z = 2 implies 1 + z = 2 implies z = 1. [Substitute the values.]
Step 8: The solution is (1, -2, 1).
Step 9: The system is consistent and independent, it has only one real solution.
Quick Summary
- Independent equations have a unique solution.
- The solution can be found using methods like substitution or elimination.
- A system of equations is independent if it is neither inconsistent nor dependent.
🍎 Teacher Insights
Emphasize the geometric interpretation of independent systems as intersecting lines/planes at a single point. Use visual aids and real-world examples to illustrate the concept.🎓 Prerequisites
- Solving linear equations
- Graphing linear inequalities
- Substitution method
- Elimination method
Check Your Knowledge
Q1: Which of the following systems of equations is independent?
Frequently Asked Questions
Q: How can I tell if a system of equations is independent?
A: If solving the system leads to a unique solution for all variables, the system is independent.
Q: What is the difference between independent, dependent, and inconsistent systems?
A: Independent systems have one unique solution, dependent systems have infinitely many solutions, and inconsistent systems have no solution.