To determine which of the equations has no solutions, we can simplify each one:
-
\(6x + 3 = -6x + 3\)
Combine like terms:
\(6x + 6x = 3 - 3\)
\(12x = 0\)
\(x = 0\)
(This has a solution: \(x = 0\)) -
\(-6x + 4 = 6x + 3\)
Combine like terms:
\(-6x - 6x = 3 - 4\)
\(-12x = -1\)
\(x = \frac{1}{12}\)
(This has a solution: \(x = \frac{1}{12}\)) -
\(6x + 4 = 6x + 4\)
This simplifies to \(0 = 0\), which is true for all values of \(x\).
(This has infinitely many solutions.) -
\(-6x + 4 = -6x + 3\)
Combine like terms:
\(-6x + 6x = 3 - 4\)
\(0 = -1\)
(This is a contradiction and has no solutions.)
The equation that has no solutions is \(-6x + 4 = -6x + 3\).