Determine whether the system is "inconsistent", "dependent" or "independent".5𝑥+2𝑦=610𝑥+4𝑦=12Group of answer choicesIndependentDependentInconsistent

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

1. Break Down the Problem

   • We have two equations:
     1. $5x + 2y = 6$ (Equation 1)
     2. $10x + 4y = 12$ (Equation 2)

2. Relevant Concepts

   • An independent system has exactly one solution.
   • A dependent system has an infinite number of solutions.
   • An inconsistent system has no solution.

3. Analysis and Detail

   • We can manipulate Equation 1 to see if Equation 2 can be derived: $\text{Multiply Equation 1 by 2:} \\ 2(5x + 2y) = 2(6) \\ 10x + 4y = 12 \text{ (which is exactly Equation 2)}$
   • Since we have found that Equation 2 can be derived by multiplying Equation 1 by 2, this means that both equations represent the same line.

4. Verify and Summarize

   • The two equations are identical in slope and intercept, confirming they are the same line.
   • Hence, there are infinitely many solutions, indicating a dependent system.

Final Answer

The system is Dependent.