Let's solve each system of equations pair by pair to match them with the appropriate solution:
1. System 1:
[tex]\[
\begin{cases}
4x - 3y = -1 \\
-3x + 4y = 6
\end{cases}
\][/tex]
This system has a unique solution which is [tex]\((2, 3)\)[/tex].
2. System 2:
[tex]\[
\begin{cases}
3x - 2y = -1 \\
-x + 2y = 3
\end{cases}
\][/tex]
This system has a unique solution which is [tex]\((1, 2)\)[/tex].
3. System 3:
[tex]\[
\begin{cases}
3x + 6y = 6 \\
2x + 4y = -4
\end{cases}
\][/tex]
This system has no solution because the lines represented by these equations are parallel.
4. System 4:
[tex]\[
\begin{cases}
-3x + 6y = -3 \\
5x - 10y = 5
\end{cases}
\][/tex]
This system has infinitely many solutions because both equations represent the same line.
Matching the systems with the appropriate solutions:
1. System:
[tex]\[
\begin{cases}
4x - 3y = -1 \\
-3x + 4y = 6
\end{cases}
\][/tex]
Solution: [tex]\((2, 3)\)[/tex].
2. System:
[tex]\[
\begin{cases}
3x - 2y = -1 \\
-x + 2y = 3
\end{cases}
\][/tex]
Solution: [tex]\((1, 2)\)[/tex].
3. System:
[tex]\[
\begin{cases}
3x + 6y = 6 \\
2x + 4y = -4
\end{cases}
\][/tex]
Solution: No solution.
4. System:
[tex]\[
\begin{cases}
-3x + 6y = -3 \\
5x - 10y = 5
\end{cases}
\][/tex]
Solution: Infinite number of solutions.
Thus, the correct matches are:
- [tex]\[
\begin{cases}
4x - 3y = -1 \\
-3x + 4y = 6
\end{cases}
\][/tex]
matched with: [tex]\((2, 3)\)[/tex].
- [tex]\[
\begin{cases}
3x - 2y = -1 \\
-x + 2y = 3
\end{cases}
\][/tex]
matched with: [tex]\((1, 2)\)[/tex].
- [tex]\[
\begin{cases}
3x + 6y = 6 \\
2x + 4y = -4
\end{cases}
\][/tex]
matched with: No solution.
- [tex]\[
\begin{cases}
-3x + 6y = -3 \\
5x - 10y = 5
\end{cases}
\][/tex]
matched with: Infinite number of solutions.
