Certainly! Let's analyze each system of equations and match them with the correct number of solutions:
1. System of equations:
[tex]\[
\begin{cases}
3x - y = 4 \\
6x - 2y = 8
\end{cases}
\][/tex]
This system has infinitely many solutions. The second equation is simply a multiple of the first equation. Hence, both equations are essentially the same line, indicating that every point on the line is a solution.
2. System of equations:
[tex]\[
\begin{cases}
y = -4x - 5 \\
y = -4x + 1
\end{cases}
\][/tex]
This system has no solution. The two equations represent parallel lines with different y-intercepts. Parallel lines do not intersect, hence there are no solutions.
3. System of equations:
[tex]\[
\begin{cases}
-3x + y = 7 \\
2x - 4y = -8
\end{cases}
\][/tex]
This system has one solution. The equations represent two lines that intersect at exactly one point. Such systems have a unique solution.
So we can complete the pairs as follows:
- Infinitely many solutions:
[tex]\[
\begin{cases}
3x - y = 4 \\
6x - 2y = 8
\end{cases}
\][/tex]
- No solution:
[tex]\[
\begin{cases}
y = -4x - 5 \\
y = -4x + 1
\end{cases}
\][/tex]
- One solution:
[tex]\[
\begin{cases}
-3x + y = 7 \\
2x - 4y = -8
\end{cases}
\][/tex]
So, the filled boxes should look like this:
Infinitely many solutions:
[tex]\[
\begin{cases}
3x - y = 4 \\
6x - 2y = 8
\end{cases}
\][/tex]
No solution:
[tex]\[
\begin{cases}
y = -4x - 5 \\
y = -4x + 1
\end{cases}
\][/tex]
One solution:
[tex]\[
\begin{cases}
-3x + y = 7 \\
2x - 4y = -8
\end{cases}
\][/tex]
