Let's go through the two equations step by step:

First Equation:

$-2(w + 2) + 7 = 3(w + 6)$

1. Distribute the $-2$ and the $3$: $-2w - 4 + 7 = 3w + 18$
2. Simplify both sides: $-2w + 3 = 3w + 18$
3. Move all terms involving $w$ to one side and constant terms to the other side: $-2w - 3w = 18 - 3$ $-5w = 15$
4. Solve for $w$: $w = \frac{15}{-5} = -3$

So, the solution is $w = -3$. The "No solution" answer chosen here is incorrect.

Second Equation:

$-6(x + 1) + 8x = 2(x - 3)$

1. Distribute the $-6$ and the $2$: $-6x - 6 + 8x = 2x - 6$
2. Simplify both sides: $2x - 6 = 2x - 6$
3. Since both sides are identical, this equation holds true for all values of $x$.

So, all real numbers are solutions to the second equation.

Final Answers:

• For the first equation, $w = -3$.
• For the second equation, all real numbers are solutions.

Would you like more details on any steps? Here are some related questions:

1. What does it mean when an equation has no solution?
2. Can an equation have multiple solutions?
3. What is the process for solving linear equations in one variable?
4. How do you verify if an equation has infinite solutions?
5. What happens when both sides of an equation simplify to the same expression?

Tip: Always simplify both sides of an equation before moving terms around—it helps avoid mistakes in handling constants and variables!