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Arrange the systems of equations in order from least to greatest based on the number of solutions for each system.

1.
[tex]\[
\begin{array}{r}
-5x + y = 10 \\
-25x + 5y = 50
\end{array}
\][/tex]

2.
[tex]\[
\begin{array}{r}
10 \\
3x - 7y = 9 \\
y = 6x - 2
\end{array}
\][/tex]

3.
[tex]\[
\begin{array}{r}
y = 6x - 4
\end{array}
\][/tex]



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]