Solve the system using any method. If the system does not have one unique solution, state whether it is inconsistent or whether the equations are dependent.

[tex]\[
\begin{array}{l}
3x - 5y = 3 \\
6x = 10y + 4
\end{array}
\][/tex]

A. The system has one solution. The solution set is { [tex]$\square$[/tex] }.
B. The system has no solution. The system is inconsistent.
C. The system has infinitely many solutions. The solution set is { [tex]$\square$[/tex] | x is any real number }.
D. The equations are dependent.



Answer :

To solve the given system of linear equations:

[tex]\[ \begin{array}{l} 3x - 5y = 3 \\ 6x = 10y + 4 \end{array} \][/tex]

we need to determine if there is a unique solution, no solution, or infinitely many solutions. Let's analyze the system step-by-step.

### Step 1: Simplify the Second Equation
First, we manipulate the second equation to a more standard form. The original equation is:

[tex]\[ 6x = 10y + 4 \][/tex]

We can rewrite it as:

[tex]\[ 6x - 10y = 4 \][/tex]

### Step 2: Consider the Equations Together
We now have the system:

[tex]\[ \begin{array}{l} 3x - 5y = 3 \\ 6x - 10y = 4 \end{array} \][/tex]

Notice that the second equation can be obtained by multiplying the first equation by 2. Specifically:

[tex]\[ 2 \cdot (3x - 5y) = 2 \cdot 3 \quad \Rightarrow \quad 6x - 10y = 6 \][/tex]

However, we have:

[tex]\[ 6x - 10y = 4 \][/tex]

This indicates a contradiction because [tex]\(6x - 10y\)[/tex] cannot be equal to both 4 and 6 at the same time.

### Step 3: Conclusion
Since we derived a contradiction from the system of equations, this means that there are no values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that can simultaneously satisfy both equations.

Thus, the system is inconsistent and has no solution.

### Final Answer
The system is inconsistent. The solution set is [tex]\(\{\}\)[/tex].