Let's analyze each pair of equations to determine which system will produce infinitely many solutions.
To have infinitely many solutions, two equations must be equivalent, meaning that one is a multiple of the other. This implies that the ratios of the coefficients of the variables and the constants should be equal.
### System 1
[tex]\[
\begin{array}{l}
2 x + 5 y = 24 \\
2 x + 5 y = 42
\end{array}
\][/tex]
Here, the equations are:
1. [tex]\( 2x + 5y = 24 \)[/tex]
2. [tex]\( 2x + 5y = 42 \)[/tex]
Comparing the ratios of the coefficients and constants:
[tex]\[
\frac{2}{2} = \frac{5}{5} \neq \frac{24}{42}
\][/tex]
The coefficients are the same, but the constants are not proportional. Therefore, this system does not have infinitely many solutions.
### System 2
[tex]\[
\begin{array}{l}
3 x - 2 y = 15 \\
6 x + 5 y = 11
\end{array}
\][/tex]
Here, the equations are:
1. [tex]\( 3x - 2y = 15 \)[/tex]
2. [tex]\( 6x + 5y = 11 \)[/tex]
The coefficients and constants are:
[tex]\[
\frac{3}{6} \neq \frac{-2}{5} \neq \frac{15}{11}
\][/tex]
The ratios are not proportional. Therefore, this system does not have infinitely many solutions.
### System 3
[tex]\[
\begin{array}{l}
4 x - 3 y = 9 \\
-8 x + 6 y = -18
\end{array}
\][/tex]
Here, the equations are:
1. [tex]\( 4x - 3y = 9 \)[/tex]
2. [tex]\( -8x + 6y = -18 \)[/tex]
Comparing the ratios:
[tex]\[
\frac{4}{-8} = \frac{-1}{2} = \frac{-3}{6} = \frac{-1}{2} = \frac{9}{-18} = \frac{1}{-2}
\][/tex]
All ratios are equal, so these equations are multiples of each other. Therefore, this system does have infinitely many solutions.
### System 4
[tex]\[
\begin{array}{l}
5 x - 3 y = 16 \\
-2 x + 3 y = -7
\end{array}
\][/tex]
Here, the equations are:
1. [tex]\( 5x - 3y = 16 \)[/tex]
2. [tex]\( -2x + 3y = -7 \)[/tex]
Comparing the ratios:
[tex]\[
\frac{5}{-2} \neq \frac{-3}{3} \neq \frac{16}{-7}
\][/tex]
The ratios are not proportional. Therefore, this system does not have infinitely many solutions.
### Conclusion
Based on the analysis, the system that produces infinitely many solutions is:
[tex]\[
\begin{array}{l}
4 x - 3 y = 9 \\
-8 x + 6 y = -18
\end{array}
\][/tex]
