Question

Which of the following equations has no solutions?(1 point) Responses 6x+3=−6x+3 6 x plus 3 equals negative 6 x plus 3 −6x+4=6x+3 negative 6 x plus 4 equals 6 x plus 3 6x+4=6x+4 6 x plus 4 equals 6 x plus 4 −6x+4=−6x+3

Answer 1

To determine which equations have no solutions, we can analyze each one step-by-step.

1. **Equation:** \( 6x + 3 = -6x + 3 \)

Solving this gives:
\[
6x + 6x = 3 - 3 \quad \Rightarrow \quad 12x = 0 \quad \Rightarrow \quad x = 0
\]
**Solution exists.**

2. **Equation:** \( -6x + 4 = 6x + 3 \)

Solving this gives:
\[
-6x - 6x = 3 - 4 \quad \Rightarrow \quad -12x = -1 \quad \Rightarrow \quad x = \frac{1}{12}
\]
**Solution exists.**

3. **Equation:** \( 6x + 4 = 6x + 4 \)

This is an identity (both sides are equal), so it is true for any value of \(x\).
**Infinitely many solutions (not no solutions).**

4. **Equation:** \( -6x + 4 = -6x + 3 \)

Rearranging gives:
\[
4 = 3
\]
This is a contradiction.
**No solution.**

After analyzing all given equations, the equation \( -6x + 4 = -6x + 3 \) is the only one that has no solutions.

So the final answer is:
**The equation with no solutions is: \(-6x + 4 = -6x + 3\)**.