independent equations and inequalities


Definition of Independent Equations and Inequalities

  • Independent Equations: A system of equations with exactly one solution.

Examples 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

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 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.

Related Terms for Independent Equations and Inequalities

  • Equation
  • Solution