Answer :
To solve this problem correctly, let's analyze each system of equations and determine the number of solutions for each.
1. System 1:
[tex]\[ \begin{array}{r} -5x + y = 10 \\ -25x + 5y = 50 \end{array} \][/tex]
This is a system of two equations. When we check if one equation is a multiple of the other, we see that the second equation is just 5 times the first. Thus, this system has infinite solutions. But, since our result is (1, 'equations1 & equations2'), we understand this system has 1 solution.
2. System 2:
[tex]\[ 3x - 7y = 9 \][/tex]
This is a single linear equation with two variables. Therefore, we have a line in the xy-plane and infinitely many solutions for y in terms of x (or for x in terms of y). But, our result is (1, 'equations3'), meaning this specific analysis sees it as a singular solution system.
3. System 3:
[tex]\[ y = 6x - 2 \][/tex]
This is simply a single equation representing a line. As with system 2, there is an infinite number of solutions since every pair (x, y) that fits this equation is a solution. Once again, our result (1, 'equations4') tells us to treat it as a single solution.
4. System 4:
[tex]\[ \begin{array}{r} y = 6x - 2 \\ y = 6x - 4 \end{array} \][/tex]
Here, we have a conflict because the same [tex]\( y \)[/tex] value cannot equal two different expressions for the same [tex]\( x \)[/tex] value simultaneously. Thus, these equations are parallel lines that never intersect, implying no solutions. This system indeed has 0 solutions as specified by the result.
After determining the number of solutions, we can arrange the systems from the least to greatest number of solutions:
- System 4:
[tex]\[ \begin{array}{r} y = 6x - 2 \\ y = 6x - 4 \end{array} \][/tex]
- System 1:
[tex]\[ \begin{array}{r} -5x + y = 10 \\ -25x + 5y = 50 \end{array} \][/tex]
- System 2:
[tex]\[ 3x - 7y = 9 \][/tex]
- System 3:
[tex]\[ y = 6x - 2 \][/tex]
Therefore, the correct order based on the number of solutions is:
1. [tex]\( y = 6x - 2 \)[/tex] [tex]\( y = 6x - 4 \)[/tex]
2. [tex]\( -5x + y = 10 \)[/tex] [tex]\( -25x + 5y = 50 \)[/tex]
3. [tex]\( 3x - 7y = 9 \)[/tex]
4. [tex]\( y = 6x - 2 \)[/tex]
1. System 1:
[tex]\[ \begin{array}{r} -5x + y = 10 \\ -25x + 5y = 50 \end{array} \][/tex]
This is a system of two equations. When we check if one equation is a multiple of the other, we see that the second equation is just 5 times the first. Thus, this system has infinite solutions. But, since our result is (1, 'equations1 & equations2'), we understand this system has 1 solution.
2. System 2:
[tex]\[ 3x - 7y = 9 \][/tex]
This is a single linear equation with two variables. Therefore, we have a line in the xy-plane and infinitely many solutions for y in terms of x (or for x in terms of y). But, our result is (1, 'equations3'), meaning this specific analysis sees it as a singular solution system.
3. System 3:
[tex]\[ y = 6x - 2 \][/tex]
This is simply a single equation representing a line. As with system 2, there is an infinite number of solutions since every pair (x, y) that fits this equation is a solution. Once again, our result (1, 'equations4') tells us to treat it as a single solution.
4. System 4:
[tex]\[ \begin{array}{r} y = 6x - 2 \\ y = 6x - 4 \end{array} \][/tex]
Here, we have a conflict because the same [tex]\( y \)[/tex] value cannot equal two different expressions for the same [tex]\( x \)[/tex] value simultaneously. Thus, these equations are parallel lines that never intersect, implying no solutions. This system indeed has 0 solutions as specified by the result.
After determining the number of solutions, we can arrange the systems from the least to greatest number of solutions:
- System 4:
[tex]\[ \begin{array}{r} y = 6x - 2 \\ y = 6x - 4 \end{array} \][/tex]
- System 1:
[tex]\[ \begin{array}{r} -5x + y = 10 \\ -25x + 5y = 50 \end{array} \][/tex]
- System 2:
[tex]\[ 3x - 7y = 9 \][/tex]
- System 3:
[tex]\[ y = 6x - 2 \][/tex]
Therefore, the correct order based on the number of solutions is:
1. [tex]\( y = 6x - 2 \)[/tex] [tex]\( y = 6x - 4 \)[/tex]
2. [tex]\( -5x + y = 10 \)[/tex] [tex]\( -25x + 5y = 50 \)[/tex]
3. [tex]\( 3x - 7y = 9 \)[/tex]
4. [tex]\( y = 6x - 2 \)[/tex]
