### Solution Process
Let's match each system of equations to the corresponding solution:
1. System of Equations:
[tex]\[
\begin{cases}
3x + y = 3 \\
-3x - 4y = 6 \\
\end{cases}
\][/tex]
Solution:
The solution for this system is [tex]\((2, -3)\)[/tex].
- [tex]\(3(2) + (-3) = 6 - 3 = 3\)[/tex]
- [tex]\(-3(2) - 4(-3) = -6 + 12 = 6\)[/tex]
2. System of Equations:
[tex]\[
\begin{cases}
4x + 12y = -8 \\
5x + 15y = -10 \\
\end{cases}
\][/tex]
Solution:
This system has an infinite number of solutions because the second equation is a multiple of the first:
[tex]\[
\begin{cases}
4x + 12y = -8 \\
\frac{5}{4} (4x + 12y) = \frac{5}{4} (-8) \\
\end{cases}
\][/tex]
Both equations describe the same line, so every solution to the first equation is also a solution to the second equation.
3. System of Equations:
[tex]\[
\begin{cases}
4x - 3y = -1 \\
2x + 3y = -5 \\
\end{cases}
\][/tex]
Solution:
The solution to this system is [tex]\((-1, -1)\)[/tex].
- [tex]\(4(-1) - 3(-1) = -4 + 3 = -1\)[/tex]
- [tex]\(2(-1) + 3(-1) = -2 - 3 = -5\)[/tex]
4. System of Equation:
[tex]\[
4x + 8y = -12
\][/tex]
Solution:
This equation has an infinite number of solutions, as it can be rearranged to form a relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
4x + 8y = -12 \implies x + 2y = -3
\][/tex]
These types of systems with a single linear equation have infinitely many solutions.
### Summary
- The system
[tex]\[
\begin{cases}
3x + y = 3 \\
-3x - 4y = 6 \\
\end{cases}
\][/tex]
has the solution [tex]\((2, -3)\)[/tex].
- The system
[tex]\[
\begin{cases}
4x + 12y = -8 \\
5x + 15y = -10 \\
\end{cases}
\][/tex]
has an infinite number of solutions.
- The system
[tex]\[
\begin{cases}
4x - 3y = -1 \\
2x + 3y = -5 \\
\end{cases}
\][/tex]
has the solution [tex]\((-1, -1)\)[/tex].
- The equation
[tex]\[
4x + 8y = -12
\][/tex]
has an infinite number of solutions.
