Find answers to your questions and expand your knowledge with IDNLearn.com. Discover in-depth and trustworthy answers to all your questions from our experienced community members.
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]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.