Answer :
We need to determine for which equations there are infinitely many solutions. An equation has infinitely many solutions when simplifying it leads to an identity (a true statement for every value of [tex]$x$[/tex]).
Let's analyze each equation step by step.
[tex]$$\textbf{Equation A: } 75x + 57 = -75x + 57$$[/tex]
1. Add [tex]$75x$[/tex] to both sides:
[tex]$$75x + 75x + 57 = 57$$[/tex]
[tex]$$150x + 57 = 57$$[/tex]
2. Subtract 57 from both sides:
[tex]$$150x = 0$$[/tex]
3. Divide by 150:
[tex]$$x = 0$$[/tex]
Since there is only one solution ([tex]$x=0$[/tex]), Equation A does not have infinitely many solutions.
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[tex]$$\textbf{Equation B: } 57x + 57 = -75x - 75$$[/tex]
1. Add [tex]$75x$[/tex] to both sides:
[tex]$$57x + 75x + 57 = -75$$[/tex]
[tex]$$132x + 57 = -75$$[/tex]
2. Subtract 57 from both sides:
[tex]$$132x = -132$$[/tex]
3. Divide by 132:
[tex]$$x = -1$$[/tex]
Since there is only one solution ([tex]$x=-1$[/tex]), Equation B does not have infinitely many solutions.
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[tex]$$\textbf{Equation C: } -75x + 57 = -75x + 57$$[/tex]
Notice that both sides of the equation are exactly the same. This means that regardless of the value of [tex]$x$[/tex], the equation is always true. Therefore, there are infinitely many solutions for Equation C.
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[tex]$$\textbf{Equation D: } -57x + 57 = -75x + 75$$[/tex]
1. Add [tex]$75x$[/tex] to both sides:
[tex]$$-57x + 75x + 57 = 75$$[/tex]
[tex]$$18x + 57 = 75$$[/tex]
2. Subtract 57 from both sides:
[tex]$$18x = 18$$[/tex]
3. Divide by 18:
[tex]$$x = 1$$[/tex]
Since there is only one solution ([tex]$x=1$[/tex]), Equation D does not have infinitely many solutions.
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After analyzing all the equations, we find that only Equation C has infinitely many solutions.
[tex]$\boxed{\text{C}}$[/tex]
Let's analyze each equation step by step.
[tex]$$\textbf{Equation A: } 75x + 57 = -75x + 57$$[/tex]
1. Add [tex]$75x$[/tex] to both sides:
[tex]$$75x + 75x + 57 = 57$$[/tex]
[tex]$$150x + 57 = 57$$[/tex]
2. Subtract 57 from both sides:
[tex]$$150x = 0$$[/tex]
3. Divide by 150:
[tex]$$x = 0$$[/tex]
Since there is only one solution ([tex]$x=0$[/tex]), Equation A does not have infinitely many solutions.
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[tex]$$\textbf{Equation B: } 57x + 57 = -75x - 75$$[/tex]
1. Add [tex]$75x$[/tex] to both sides:
[tex]$$57x + 75x + 57 = -75$$[/tex]
[tex]$$132x + 57 = -75$$[/tex]
2. Subtract 57 from both sides:
[tex]$$132x = -132$$[/tex]
3. Divide by 132:
[tex]$$x = -1$$[/tex]
Since there is only one solution ([tex]$x=-1$[/tex]), Equation B does not have infinitely many solutions.
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[tex]$$\textbf{Equation C: } -75x + 57 = -75x + 57$$[/tex]
Notice that both sides of the equation are exactly the same. This means that regardless of the value of [tex]$x$[/tex], the equation is always true. Therefore, there are infinitely many solutions for Equation C.
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[tex]$$\textbf{Equation D: } -57x + 57 = -75x + 75$$[/tex]
1. Add [tex]$75x$[/tex] to both sides:
[tex]$$-57x + 75x + 57 = 75$$[/tex]
[tex]$$18x + 57 = 75$$[/tex]
2. Subtract 57 from both sides:
[tex]$$18x = 18$$[/tex]
3. Divide by 18:
[tex]$$x = 1$$[/tex]
Since there is only one solution ([tex]$x=1$[/tex]), Equation D does not have infinitely many solutions.
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After analyzing all the equations, we find that only Equation C has infinitely many solutions.
[tex]$\boxed{\text{C}}$[/tex]
