Answer :
To determine which equations have infinitely many solutions, we need to understand when an equation has such solutions. An equation has infinitely many solutions if, and only if, the left-hand side (LHS) is identical to the right-hand side (RHS).
Let's examine each option step-by-step.
### Option A: [tex]\( -75x + 57 = -75x + 57 \)[/tex]
Here, the left-hand side is [tex]\( -75x + 57 \)[/tex] and the right-hand side is also [tex]\( -75x + 57 \)[/tex]:
[tex]\[ -75x + 57 = -75x + 57 \][/tex]
Both sides of the equation are identical. Therefore, this equation is true for all values of [tex]\( x \)[/tex] and has infinitely many solutions.
### Option B: [tex]\( 75x + 57 = -75x + 57 \)[/tex]
For this equation, the left-hand side is [tex]\( 75x + 57 \)[/tex] and the right-hand side is [tex]\( -75x + 57 \)[/tex]:
[tex]\[ 75x + 57 = -75x + 57 \][/tex]
Let's simplify and see if these sides can ever be equal:
[tex]\[ 75x + 57 = -75x + 57 \][/tex]
Subtract [tex]\( 57 \)[/tex] from both sides:
[tex]\[ 75x = -75x \][/tex]
Add [tex]\( 75x \)[/tex] to both sides:
[tex]\[ 150x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
This equation has a single solution [tex]\( x = 0 \)[/tex].
### Option C: [tex]\( -57x + 57 = -75x + 75 \)[/tex]
Here, the left-hand side is [tex]\( -57x + 57 \)[/tex] and the right-hand side is [tex]\( -75x + 75 \)[/tex]:
[tex]\[ -57x + 57 = -75x + 75 \][/tex]
Simplify and compare both sides:
[tex]\[ -57x + 57 = -75x + 75 \][/tex]
Add [tex]\( 75x \)[/tex] to both sides:
[tex]\[ 18x + 57 = 75 \][/tex]
Subtract [tex]\( 57 \)[/tex] from both sides:
[tex]\[ 18x = 18 \][/tex]
[tex]\[ x = 1 \][/tex]
This equation has a single solution [tex]\( x = 1 \)[/tex].
### Option D: [tex]\( 57x + 57 = -75x - 75 \)[/tex]
For this equation, the left-hand side is [tex]\( 57x + 57 \)[/tex] and the right-hand side is [tex]\( -75x - 75 \)[/tex]:
[tex]\[ 57x + 57 = -75x - 75 \][/tex]
Simplify and compare both sides:
[tex]\[ 57x + 57 = -75x - 75 \][/tex]
Add [tex]\( 75x \)[/tex] to both sides:
[tex]\[ 132x + 57 = -75 \][/tex]
Subtract [tex]\( 57 \)[/tex] from both sides:
[tex]\[ 132x = -132 \][/tex]
[tex]\[ x = -1 \][/tex]
This equation has a single solution [tex]\( x = -1 \)[/tex].
### Conclusion
Only option A [tex]\( -75x + 57 = -75x + 57 \)[/tex] has infinitely many solutions because the left-hand side is identical to the right-hand side. The other options have only a single solution each.
Let's examine each option step-by-step.
### Option A: [tex]\( -75x + 57 = -75x + 57 \)[/tex]
Here, the left-hand side is [tex]\( -75x + 57 \)[/tex] and the right-hand side is also [tex]\( -75x + 57 \)[/tex]:
[tex]\[ -75x + 57 = -75x + 57 \][/tex]
Both sides of the equation are identical. Therefore, this equation is true for all values of [tex]\( x \)[/tex] and has infinitely many solutions.
### Option B: [tex]\( 75x + 57 = -75x + 57 \)[/tex]
For this equation, the left-hand side is [tex]\( 75x + 57 \)[/tex] and the right-hand side is [tex]\( -75x + 57 \)[/tex]:
[tex]\[ 75x + 57 = -75x + 57 \][/tex]
Let's simplify and see if these sides can ever be equal:
[tex]\[ 75x + 57 = -75x + 57 \][/tex]
Subtract [tex]\( 57 \)[/tex] from both sides:
[tex]\[ 75x = -75x \][/tex]
Add [tex]\( 75x \)[/tex] to both sides:
[tex]\[ 150x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
This equation has a single solution [tex]\( x = 0 \)[/tex].
### Option C: [tex]\( -57x + 57 = -75x + 75 \)[/tex]
Here, the left-hand side is [tex]\( -57x + 57 \)[/tex] and the right-hand side is [tex]\( -75x + 75 \)[/tex]:
[tex]\[ -57x + 57 = -75x + 75 \][/tex]
Simplify and compare both sides:
[tex]\[ -57x + 57 = -75x + 75 \][/tex]
Add [tex]\( 75x \)[/tex] to both sides:
[tex]\[ 18x + 57 = 75 \][/tex]
Subtract [tex]\( 57 \)[/tex] from both sides:
[tex]\[ 18x = 18 \][/tex]
[tex]\[ x = 1 \][/tex]
This equation has a single solution [tex]\( x = 1 \)[/tex].
### Option D: [tex]\( 57x + 57 = -75x - 75 \)[/tex]
For this equation, the left-hand side is [tex]\( 57x + 57 \)[/tex] and the right-hand side is [tex]\( -75x - 75 \)[/tex]:
[tex]\[ 57x + 57 = -75x - 75 \][/tex]
Simplify and compare both sides:
[tex]\[ 57x + 57 = -75x - 75 \][/tex]
Add [tex]\( 75x \)[/tex] to both sides:
[tex]\[ 132x + 57 = -75 \][/tex]
Subtract [tex]\( 57 \)[/tex] from both sides:
[tex]\[ 132x = -132 \][/tex]
[tex]\[ x = -1 \][/tex]
This equation has a single solution [tex]\( x = -1 \)[/tex].
### Conclusion
Only option A [tex]\( -75x + 57 = -75x + 57 \)[/tex] has infinitely many solutions because the left-hand side is identical to the right-hand side. The other options have only a single solution each.
