Answer :
Sure, let's analyze each of the given equations to determine if they have infinitely many solutions:
### Equation (A): [tex]$75x + 57 = -75x + 57$[/tex]
To check if this equation has infinitely many solutions, let's simplify it.
1. Subtract [tex]$-75x$[/tex] from 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]
This equation has a unique solution [tex]\( x = 0 \)[/tex]. Therefore, it does not have infinitely many solutions.
### Equation (B): [tex]$57x + 57 = -75x - 75$[/tex]
To check if this equation has infinitely many solutions, let's simplify it.
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]
This equation has a unique solution [tex]\( x = -1 \)[/tex]. Therefore, it does not have infinitely many solutions.
### Equation (C): [tex]$-75x + 57 = -75x + 57$[/tex]
To check if this equation has infinitely many solutions, let's simplify it.
1. Notice that both sides of the equation are already identical:
[tex]\[ -75x + 57 = -75x + 57 \][/tex]
Since both sides are the same for all values of [tex]\( x \)[/tex], this equation is always true for any value of [tex]\( x \)[/tex]. Therefore, it does have infinitely many solutions.
### Equation (D): [tex]$-57x + 57 = -75x + 75$[/tex]
To check if this equation has infinitely many solutions, let's simplify it.
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]
This equation has a unique solution [tex]\( x = 1 \)[/tex]. Therefore, it does not have infinitely many solutions.
### Conclusion
The only equation that has infinitely many solutions is:
[tex]\[ \boxed{3} \][/tex]
Thus, the answer is equation (C) [tex]\( -75x + 57 = -75x + 57 \)[/tex].
### Equation (A): [tex]$75x + 57 = -75x + 57$[/tex]
To check if this equation has infinitely many solutions, let's simplify it.
1. Subtract [tex]$-75x$[/tex] from 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]
This equation has a unique solution [tex]\( x = 0 \)[/tex]. Therefore, it does not have infinitely many solutions.
### Equation (B): [tex]$57x + 57 = -75x - 75$[/tex]
To check if this equation has infinitely many solutions, let's simplify it.
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]
This equation has a unique solution [tex]\( x = -1 \)[/tex]. Therefore, it does not have infinitely many solutions.
### Equation (C): [tex]$-75x + 57 = -75x + 57$[/tex]
To check if this equation has infinitely many solutions, let's simplify it.
1. Notice that both sides of the equation are already identical:
[tex]\[ -75x + 57 = -75x + 57 \][/tex]
Since both sides are the same for all values of [tex]\( x \)[/tex], this equation is always true for any value of [tex]\( x \)[/tex]. Therefore, it does have infinitely many solutions.
### Equation (D): [tex]$-57x + 57 = -75x + 75$[/tex]
To check if this equation has infinitely many solutions, let's simplify it.
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]
This equation has a unique solution [tex]\( x = 1 \)[/tex]. Therefore, it does not have infinitely many solutions.
### Conclusion
The only equation that has infinitely many solutions is:
[tex]\[ \boxed{3} \][/tex]
Thus, the answer is equation (C) [tex]\( -75x + 57 = -75x + 57 \)[/tex].
