Question: 11 - 6c + 3(4 + c) = 5(5 - c) - 3c
1. **State the problem:** Solve the equation $$11 - 6c + 3(4 + c) = 5(5 - c) - 3c$$ for the variable $$c$$.
2. **Apply the distributive property:** Expand the terms with parentheses.
$$11 - 6c + 3 \times 4 + 3 \times c = 5 \times 5 - 5 \times c - 3c$$
which simplifies to
$$11 - 6c + 12 + 3c = 25 - 5c - 3c$$
3. **Combine like terms on each side:**
Left side: $$11 + 12 = 23$$ and $$-6c + 3c = -3c$$
Right side: $$-5c - 3c = -8c$$
So the equation becomes
$$23 - 3c = 25 - 8c$$
4. **Isolate variable terms on one side:** Add $$8c$$ to both sides to move all $$c$$ terms to the left.
$$23 - 3c + 8c = 25 - 8c + 8c$$
which simplifies to
$$23 + 5c = 25$$
5. **Isolate the constant term:** Subtract $$23$$ from both sides.
$$23 + 5c - 23 = 25 - 23$$
which simplifies to
$$5c = 2$$
6. **Solve for $$c$$:** Divide both sides by $$5$$.
$$\frac{\cancel{5}c}{\cancel{5}} = \frac{2}{5}$$
which gives
$$c = \frac{2}{5}$$
**Final answer:** $$c = \frac{2}{5}$$