Determine whether the system is "inconsistent", "dependent" or "independent".3𝑥−4𝑦=312𝑥−16𝑦=8 Group of answer choicesDependentIndependentInconsistent

To determine whether the system of equations is "inconsistent," "dependent," or "independent," we follow these steps:

1. Break Down the Problem

We have the following system of equations:

1. $3x - 4y = 3$ (Equation 1)
2. $12x - 16y = 8$ (Equation 2)

We need to analyze these equations for their relationships with one another.

2. Relevant Concepts

• Inconsistent: No solution exists (the lines represented by the equations do not intersect).
• Dependent: Infinitely many solutions exist (the lines represented by the equations are the same).
• Independent: Exactly one solution exists (the lines intersect at one point).

3. Analysis and Detail

Let's simplify Equation 2 to see its relationship to Equation 1.

• Divide Equation 2 by 4: $\frac{12x - 16y = 8}{4} \implies 3x - 4y = 2$ This gives us a new form of Equation 2: $3x - 4y = 2 \quad \text{(Equation 2')}$

Now, we can rewrite both equations for comparison:

1. $3x - 4y = 3$ (Equation 1)
2. $3x - 4y = 2$ (Equation 2')

Clearly, both equations represent lines with the same slope but different y-intercepts. Thus, they are parallel and never intersect.

4. Verify and Summarize

Since the equations represent parallel lines, they do not have any point in common; therefore, they do not intersect. This indicates the equations are inconsistent.

Final Answer

The system of equations is Inconsistent.