To determine which of the given equations have no solutions, we need to carefully analyze each equation step-by-step. Specifically, we examine if there is a contradiction in the equality.
Let's go through each one:
### Equation (A)
[tex]\[ 33x - 25 = 33x + 25 \][/tex]
1. Subtract [tex]\( 33x \)[/tex] from both sides:
[tex]\[ 33x - 25 - 33x = 33x + 25 - 33x \][/tex]
Simplifies to:
[tex]\[ -25 = 25 \][/tex]
This is a contradiction because [tex]\(-25\)[/tex] does not equal [tex]\(25\)[/tex]. Therefore, this equation has no solutions.
### Equation (B)
[tex]\[ 33x - 33 = 33x + 25 \][/tex]
1. Subtract [tex]\( 33x \)[/tex] from both sides:
[tex]\[ 33x - 33 - 33x = 33x + 25 - 33x \][/tex]
Simplifies to:
[tex]\[ -33 = 25 \][/tex]
This is a contradiction because [tex]\(-33\)[/tex] does not equal [tex]\(25\)[/tex]. Therefore, this equation has no solutions.
### Equation (C)
[tex]\[ 33x + 25 = 33x + 25 \][/tex]
1. Subtract [tex]\( 33x \)[/tex] from both sides:
[tex]\[ 33x + 25 - 33x = 33x + 25 - 33x \][/tex]
Simplifies to:
[tex]\[ 25 = 25 \][/tex]
This is a true statement. It means that the equation is always true for any value of [tex]\( x \)[/tex], indicating that it has infinitely many solutions.
### Equation (D)
[tex]\[ 33x + 33 = 33x + 25 \][/tex]
1. Subtract [tex]\( 33x \)[/tex] from both sides:
[tex]\[ 33x + 33 - 33x = 33x + 25 - 33x \][/tex]
Simplifies to:
[tex]\[ 33 = 25 \][/tex]
This is a contradiction because [tex]\(33\)[/tex] does not equal [tex]\(25\)[/tex]. Therefore, this equation has no solutions.
### Summary
Based on our analysis, the equations that have no solutions are:
- (A) [tex]\( 33x - 25 = 33x + 25 \)[/tex]
- (B) [tex]\( 33x - 33 = 33x + 25 \)[/tex]
- (D) [tex]\( 33x + 33 = 33x + 25 \)[/tex]
Therefore, the correct answers are:
[tex]\[ \boxed{1, 2, 4} \][/tex]
