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Determine which system will produce infinitely many solutions.

A.
[tex]\[
\begin{array}{l}
2x + 5y = 24 \\
2x + 5y = 42
\end{array}
\][/tex]

B.
[tex]\[
3x - 2y = 15 \\
6x + 5y = 11
\][/tex]

C.
[tex]\[
\begin{array}{l}
4x - 3y = 9 \\
-8x + 6y = -18
\end{array}
\][/tex]

D.
[tex]\[
\begin{array}{l}
5x - 3y = 16 \\
-2x + 3y = -7
\end{array}
\][/tex]


Sagot :

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]