Question: Determine which equations have the same solution set as $$\frac{2}{3} - x + \frac{1}{6} = 6x$$ by recognizing properties, rather than solving. Check all that apply.
$$4 - 6x + 1 = 36x$$
$$\frac{5}{6} - x = 6x$$
$$4 - x + 1 = 6x$$
$$\frac{5}{6} + x = 6x$$
$$5 = 30x$$
$$5 = 42x$$
1. **State the problem:** We want to find which of the given equations have the same solution set as the equation $$\frac{2}{3} - x + \frac{1}{6} = 6x$$ without solving them directly.
2. **Simplify the original equation:** Combine the fractions on the left side.
$$\frac{2}{3} + \frac{1}{6} - x = 6x$$
Find a common denominator for $$\frac{2}{3}$$ and $$\frac{1}{6}$$ which is 6:
$$\frac{4}{6} + \frac{1}{6} - x = 6x$$
$$\frac{5}{6} - x = 6x$$
3. **Rewrite the original equation:**
$$\frac{5}{6} - x = 6x$$
4. **Compare each given equation to the simplified original:**
- $$4 - 6x + 1 = 36x$$ simplifies to $$5 - 6x = 36x$$ which is not equivalent to $$\frac{5}{6} - x = 6x$$.
- $$\frac{5}{6} - x = 6x$$ is exactly the simplified original equation, so it has the same solution set.
- $$4 - x + 1 = 6x$$ simplifies to $$5 - x = 6x$$ which is not equivalent to the original.
- $$\frac{5}{6} + x = 6x$$ differs by the sign of $$x$$ on the left side, so it is not equivalent.
- $$5 = 30x$$ can be rewritten as $$\frac{5}{6} = x$$ after dividing both sides by 6, but this is not the same as the original equation's solution.
- $$5 = 42x$$ is different and not equivalent.
5. **Conclusion:** Only the equation $$\frac{5}{6} - x = 6x$$ has the same solution set as the original equation.
**Final answer:**
Only $$\frac{5}{6} - x = 6x$$ matches the original equation's solution set.