To solve for which system of equations led to the intermediate step of [tex]\(5x = 20\)[/tex], we need to analyze the given systems and apply the elimination method.
### System 1
1. [tex]\(3x + 3y = 24\)[/tex]
2. [tex]\(-2x - 3y = 4\)[/tex]
Add the two equations:
[tex]\[ (3x + 3y) + (-2x - 3y) = 24 + 4 \][/tex]
[tex]\[ 3x + 3y - 2x - 3y = 28 \][/tex]
[tex]\[ x = 28 \][/tex]
This does not lead to [tex]\(5x = 20\)[/tex].
### System 2
1. [tex]\(6x - y = 15\)[/tex]
2. [tex]\(x + y = 5\)[/tex]
Add the two equations:
[tex]\[ (6x - y) + (x + y) = 15 + 5 \][/tex]
[tex]\[ 6x - y + x + y = 20 \][/tex]
[tex]\[ 7x = 20 \][/tex]
[tex]\[ x = \frac{20}{7} \][/tex]
This does not lead to [tex]\(5x = 20\)[/tex].
### System 3
1. [tex]\(3x - 2y = 10\)[/tex]
2. [tex]\(5x + 2y = 20\)[/tex]
Add the two equations:
[tex]\[ (3x - 2y) + (5x + 2y) = 10 + 20 \][/tex]
[tex]\[ 3x - 2y + 5x + 2y = 30 \][/tex]
[tex]\[ 8x = 30 \][/tex]
[tex]\[ x = \frac{30}{8} \][/tex]
This does not lead to [tex]\(5x = 20\)[/tex].
### System 4
1. [tex]\(5x + 2y = 20\)[/tex]
2. [tex]\(-5x - 2y = -20\)[/tex]
Add the two equations:
[tex]\[ (5x + 2y) + (-5x - 2y) = 20 + (-20) \][/tex]
[tex]\[ 5x + 2y - 5x - 2y = 0 \][/tex]
[tex]\[ 0 = 0 \][/tex]
This shows that the two equations are essentially the same, leading to a tautology. Thus, they can indeed yield an intermediate step of [tex]\(5x = 20\)[/tex] in the process of elimination.
Thus, the system which could have led to the intermediate equation [tex]\(5x = 20\)[/tex] is:
1. [tex]\(5x + 2y = 20\)[/tex]
2. [tex]\(-5x - 2y = -20\)[/tex]
This matches the criterion since summing these equations makes it evident that they align with the intermediate step of the solution. The correct choice is the system containing the equations:
[tex]\[ \boxed{5x + 2y = 20 \text{ and } -5x - 2y = -20} \][/tex]
