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:

     example of   Independent Equations and Inequalities
    Choices:
    A. Inconsistent
    B. Consistent and independent
    C. Consistent and dependent
    D. Consistent
    Correct Answer: B

Solution:

    Step 1: [Multiply the third equation by then,-1/2add this equation to the first equation.]
       example of   Independent Equations and Inequalities
    Step 2: y = - 2 [Solve for y.]
    Step 3:  example of   Independent Equations and Inequalities [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.