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].
[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].
