Connect with knowledgeable individuals and find the best answers at IDNLearn.com. Our platform is designed to provide quick and accurate answers to any questions you may have.
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]
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]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.