To determine which of the given equations has no solutions, let's analyze each one:
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Equation: \( 6x + 4 = 6x + 3 \)
- Subtract \( 6x \) from both sides:
- \( 4 = 3 \) (This is false, hence no solutions.)
-
Equation: \( 6x + 3 = -6x + 3 \)
- Add \( 6x \) to both sides:
- \( 12x + 3 = 3 \)
- Subtract 3 from both sides:
- \( 12x = 0 \) ⇒ \( x = 0 \) (This has a solution.)
-
Equation: \( 6x + 4 = 6x + 4 \)
- Subtract \( 6x + 4 \) from both sides:
- \( 0 = 0 \) (This is always true, hence infinitely many solutions.)
-
Equation: \( -6x + 4 = -6x + 3 \)
- Add \( 6x \) to both sides:
- \( 4 = 3 \) (This is false, hence no solutions.)
From the above analysis, the equations that have no solutions are:
- \( 6x + 4 = 6x + 3 \)
- \( -6x + 4 = -6x + 3 \)
Thus, both \( 6x + 4 = 6x + 3 \) and \( -6x + 4 = -6x + 3 \) have no solutions. However, since you are looking for a single equation, either of those can be the correct answer depending on which one you want to highlight.
The two equations with no solutions are:
- \( 6x + 4 = 6x + 3 \) (also indicated, but just one is needed)
- \( -6x + 4 = -6x + 3 \)
Choose one of them based on your required answer.