# 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* + 3*z* = 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:** 5*x* + 5*z* = 10 [Multiply the third equation by 5.]

**Step 5:**

2*x* - 5*z* = - 3

5*x* + 5*z* = 10

7*x* = 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 **