﻿ Definition and examples independent equations and inequalities | define independent equations and inequalities - Free Math Dictionary Online

# Independent Equations and Inequalities

## Definition of Independent Equations and Inequalities

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

### 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

### 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
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.

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