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Definition of Independent Equations and Inequalities

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

Examples of Independent

• The system of equations given below is independent.

x + y + 3z = 12

y + z = - 4

z = 2

Solved Example on Independent

State whether the system is consistent and independent, consistent and dependent, or inconsistent:

Choices:

A. Inconsistent
B. Consistent and independent
C. Consistent and dependent

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

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.

• Equation
• Solution