Certainly! To classify these systems of equations, we need to determine whether each pair has a single solution, no solution, or infinite solutions. Let’s go through each pair step by step:
### Pair 1:
[tex]\[
\begin{cases}
y = 5 - 2x \\
x = 26 - 3y
\end{cases}
\][/tex]
This system has infinite solutions.
### Pair 2:
[tex]\[
\begin{cases}
5x + 4y = 6 \\
x + 2y = 3
\end{cases}
\][/tex]
This system also has infinite solutions.
### Pair 3:
[tex]\[
\begin{cases}
4x + 2y = 10 \\
2x + 6y = 22
\end{cases}
\][/tex]
This system has infinite solutions as well.
### Pair 4:
[tex]\[
\begin{cases}
10x - 2y = 7 \\
4x + 8y = 15
\end{cases}
\][/tex]
This system also has infinite solutions.
### Pair 5:
[tex]\[
\begin{cases}
3x + 4y = 17 \\
-6x = 10y - 39
\end{cases}
\][/tex]
Again, this system has infinite solutions.
### Pair 6:
[tex]\[
\begin{cases}
x + 5y = 24 \\
5x = 12 - y
\end{cases}
\][/tex]
This system too has infinite solutions.
The classification table based on the above analysis is:
[tex]\[
\begin{array}{|l|l|l|}
\hline
\text{Single Solution} & \text{No Solution} & \text{Infinite Solutions} \\
\hline
& & \\
\hline
& & \\
\hline
& & \inf \\
\hline
& & \inf \\
\hline
& & \inf \\
\hline
& & \inf \\
\hline
& & \inf \\
\hline
& & \inf \\
\hline
\end{array}
\][/tex]
