Let's analyze the given system of linear equations and deduce the key points:
We are given the system:
[tex]\[ -2x + y = -6 \][/tex]
[tex]\[ 2x = -6 + y \][/tex]
Firstly, we want to determine the relationship between the two equations. Let's rewrite both in standard form.
Equation 1:
[tex]\[ -2x + y = -6 \][/tex]
Equation 2 can be rearranged as:
[tex]\[ 2x - y = -6 \][/tex]
### Step-by-Step Solution:
1. Set up the Equations:
[tex]\[ -2x + y = -6 \ \ \ \ \ \text{(Equation 1)}\][/tex]
[tex]\[ 2x - y = -6 \ \ \ \ \ \text{(Equation 2) }\][/tex]
2. Add the Equations:
By adding Equation 1 and Equation 2, we eliminate [tex]\(y\)[/tex]:
[tex]\[
(-2x + y) + (2x - y) = -6 + (-6)
\][/tex]
[tex]\[
-2x + y + 2x - y = -12
\][/tex]
[tex]\[
0 = -12
\][/tex]
This equation, [tex]\(0 = -12\)[/tex], is a contradiction.
3. Conclusion about the Nature of the Solutions:
Since we've arrived at a contradiction, it implies that there is no solution that fits both equations simultaneously. Therefore, the system has no solution.
4. Interpreting the Results:
[tex]\[
\begin{tabular}{|c|l}
\hline
System B & The system has no solution. \\
$(x, y) = (\square, \square)$ \\
$-2x + y = -6$ & The system has a unique solution: \\
$2x = -6 + y$ & The system has infinitely many solutions. \\
They must satisfy the following equation: \\
$y = \square$ \\
\end{tabular}
\][/tex]
From our analysis:
- The system has no solution: True.
- The system has a unique solution: False.
- The system has infinitely many solutions: False.
- Specific equations satisfying infinite solutions: Not applicable since there are no solutions.
Therefore, we fill in the table as follows:
[tex]\[
\begin{tabular}{|c|l}
\hline
System B & The system has no solution. \\
$(x, y) = (\square, \square)$ \\
$-2x + y = -6$ & The system has a unique solution: \\
$2x = -6 + y$ & The system has infinitely many solutions. \\
They must satisfy the following equation: \\
$y = \square$ \\
\end{tabular}
\][/tex]
Given the nature of the system:
- [tex]$(x,y) = (\square, \square)$[/tex] remains blank indicating no solution exists.
- The system has no solution: Checked as true.
