Answer :
We need to determine for which equation the equality holds for every possible real value of [tex]$x$[/tex]. An equation with infinitely many solutions is an identity, meaning after simplifying both sides the result is always true (typically reducing to [tex]$0=0$[/tex]).
Let’s examine each equation.
________________________________________
Equation A:
[tex]$$75x + 57 = -75x + 57$$[/tex]
Step 1. Bring all terms to one side:
[tex]\[
75x + 57 - (-75x + 57) = 75x + 57 + 75x - 57 = 150x
\][/tex]
Step 2. Set the simplified expression equal to 0:
[tex]\[
150x = 0 \quad \Longrightarrow \quad x = 0
\][/tex]
Since there is only one solution ([tex]$x = 0$[/tex]), Equation A does not have infinitely many solutions.
________________________________________
Equation B:
[tex]$$57x + 57 = -75x - 75$$[/tex]
Step 1. Bring all terms to one side:
[tex]\[
57x + 57 - (-75x - 75) = 57x + 57 + 75x + 75 = 132x + 132
\][/tex]
Step 2. Set the simplified expression equal to 0:
[tex]\[
132x + 132 = 0 \quad \Longrightarrow \quad 132x = -132 \quad \Longrightarrow \quad x = -1
\][/tex]
Since there is only one solution ([tex]$x = -1$[/tex]), Equation B does not have infinitely many solutions.
________________________________________
Equation C:
[tex]$$-75x + 57 = -75x + 57$$[/tex]
Step 1. Bring all terms to one side:
[tex]\[
-75x + 57 - (-75x + 57) = -75x + 57 + 75x - 57 = 0
\][/tex]
The simplified expression is [tex]$0$[/tex], which is true for every [tex]$x$[/tex]. Therefore, Equation C is an identity and has infinitely many solutions.
________________________________________
Equation D:
[tex]$$-57x + 57 = -75x + 75$$[/tex]
Step 1. Bring all terms to one side:
[tex]\[
-57x + 57 - (-75x + 75) = -57x + 57 + 75x - 75 = 18x - 18
\][/tex]
Step 2. Set the simplified expression equal to 0:
[tex]\[
18x - 18 = 0 \quad \Longrightarrow \quad 18x = 18 \quad \Longrightarrow \quad x = 1
\][/tex]
Since there is only one solution ([tex]$x = 1$[/tex]), Equation D does not have infinitely many solutions.
________________________________________
Conclusion:
The only equation that holds for every value of [tex]$x$[/tex] (i.e., is an identity) is Equation C. Therefore, the answer is:
[tex]$$\textbf{C}$$[/tex]
Let’s examine each equation.
________________________________________
Equation A:
[tex]$$75x + 57 = -75x + 57$$[/tex]
Step 1. Bring all terms to one side:
[tex]\[
75x + 57 - (-75x + 57) = 75x + 57 + 75x - 57 = 150x
\][/tex]
Step 2. Set the simplified expression equal to 0:
[tex]\[
150x = 0 \quad \Longrightarrow \quad x = 0
\][/tex]
Since there is only one solution ([tex]$x = 0$[/tex]), Equation A does not have infinitely many solutions.
________________________________________
Equation B:
[tex]$$57x + 57 = -75x - 75$$[/tex]
Step 1. Bring all terms to one side:
[tex]\[
57x + 57 - (-75x - 75) = 57x + 57 + 75x + 75 = 132x + 132
\][/tex]
Step 2. Set the simplified expression equal to 0:
[tex]\[
132x + 132 = 0 \quad \Longrightarrow \quad 132x = -132 \quad \Longrightarrow \quad x = -1
\][/tex]
Since there is only one solution ([tex]$x = -1$[/tex]), Equation B does not have infinitely many solutions.
________________________________________
Equation C:
[tex]$$-75x + 57 = -75x + 57$$[/tex]
Step 1. Bring all terms to one side:
[tex]\[
-75x + 57 - (-75x + 57) = -75x + 57 + 75x - 57 = 0
\][/tex]
The simplified expression is [tex]$0$[/tex], which is true for every [tex]$x$[/tex]. Therefore, Equation C is an identity and has infinitely many solutions.
________________________________________
Equation D:
[tex]$$-57x + 57 = -75x + 75$$[/tex]
Step 1. Bring all terms to one side:
[tex]\[
-57x + 57 - (-75x + 75) = -57x + 57 + 75x - 75 = 18x - 18
\][/tex]
Step 2. Set the simplified expression equal to 0:
[tex]\[
18x - 18 = 0 \quad \Longrightarrow \quad 18x = 18 \quad \Longrightarrow \quad x = 1
\][/tex]
Since there is only one solution ([tex]$x = 1$[/tex]), Equation D does not have infinitely many solutions.
________________________________________
Conclusion:
The only equation that holds for every value of [tex]$x$[/tex] (i.e., is an identity) is Equation C. Therefore, the answer is:
[tex]$$\textbf{C}$$[/tex]
