Determine whether the system is "inconsistent", "dependent" or "independent".3𝑥+4𝑦=9−6𝑥−8𝑦=10Group of answer choicesInconsistentDependentIndependent

To determine whether the given system of equations is inconsistent, dependent, or independent, we can analyze the equations:

1. Given Equations: $3x + 4y = 9 \quad \text{(1)}$ $-6x - 8y = 10 \quad \text{(2)}$

2. Break Down the Problem:

   • We will first manipulate Equation (2) to see how it relates to Equation (1).

3. Relevant Concepts:

   • A system of linear equations can be classified as:
     • Independent: One unique solution exists.
     • Dependent: Infinite solutions exist (the equations represent the same line).
     • Inconsistent: No solution exists (the lines are parallel).

4. Analysis and Detail:

   • Let's manipulate Equation (2) by dividing all terms by -2: $3x + 4y = -5 \quad \text{(3)}$
   • Now we compare Equation (1) and Equation (3): $3x + 4y = 9 \quad \text{(1)}$ $3x + 4y = -5 \quad \text{(3)}$
   • Since the left-hand sides of Equations (1) and (3) are the same, but the right-hand sides are not equal ($9 \neq -5$), the two equations represent parallel lines.

5. Verify and Summarize:

   • Parallel lines do not intersect, which means there is no solution to this system of equations.
   • Therefore, the system is inconsistent.

Final Answer

The system is inconsistent.