Let's analyze the errors in solving each equation provided.

a) Given equation:

$3x - 7 = -2x + 8$

Work shown:

1. $3x + (-2x) = 8 + 7$
2. $x = 15$

Error:

The mistake occurs in step 1. When moving the $-7$ to the right side, it should be added to $8$, not swapped with its sign. Here's the correct approach:

Correct Solution:

1. Start with $3x - 7 = -2x + 8$.
2. Add $2x$ to both sides: $3x + 2x - 7 = 8$, which simplifies to: $5x - 7 = 8$
3. Add 7 to both sides: $5x = 8 + 7$, which gives: $5x = 15$
4. Divide by 5: $x = 3$.

b) Given equation:

$2(v - 5) = -(3v + 5)$

Work shown:

1. $2v - 10 = -3v - 5$
2. $5v = 15$
3. $v = 3$

Error:

The steps seem correct, but there's an issue with distributing the negative sign on the right-hand side of the original equation. Let's work through it carefully:

Correct Solution:

1. Start with $2(v - 5) = -(3v + 5)$.
2. Distribute both sides: $2v - 10 = -3v - 5$
3. Add $3v$ to both sides: $2v + 3v - 10 = -5$ Simplifying, we get: $5v - 10 = -5$
4. Add 10 to both sides: $5v = 5$
5. Divide by 5: $v = 1$.

Would you like more details or have any questions on these corrections?

Here are some related questions to explore:

1. How do you determine when to add or subtract terms while solving an equation?
2. What steps do you take to correctly distribute a negative sign across parentheses?
3. How can combining like terms on both sides of an equation simplify your work?
4. What is the importance of checking your solution after solving an equation?
5. Can you explain the difference between transposing terms and combining like terms?

Tip: Always double-check the distribution of negative signs when simplifying equations!