Let's go through each of the problems one by one.

1) $-\frac{3}{x} = 7$

We want to solve for $x$.

First, multiply both sides by $x$ to eliminate the denominator:

$-3 = 7x$

Now, divide both sides by 7:

$x = \frac{-3}{7}$

So, the solution is:

$x = -\frac{3}{7}$

2) $4(x + 3) = 2x + 2$

Distribute the 4 on the left-hand side:

$4x + 12 = 2x + 2$

Now, subtract $2x$ from both sides:

$2x + 12 = 2$

Next, subtract 12 from both sides:

$2x = -10$

Finally, divide both sides by 2:

$x = -5$

So, the solution is:

$x = -5$

3) $10x + 4.5 = 2x$

First, subtract $2x$ from both sides:

$8x + 4.5 = 0$

Now, subtract 4.5 from both sides:

$8x = -4.5$

Finally, divide both sides by 8:

$x = \frac{-4.5}{8} = -0.5625$

So, the solution is:

$x = -0.5625$

4) $2(x + 7) = 2x + 10 + 4$

First, distribute the 2 on the left-hand side:

$2x + 14 = 2x + 14$

This simplifies to $2x + 14 = 2x + 14$, which is always true.

Part (a) "identity or contradiction?"

Since the equation is always true, it is an identity.

Part (b) "no solution or many solutions?"

Since this is an identity, the equation has many solutions (specifically, infinitely many solutions).

Let me know if you'd like more details on any of the solutions! Here are 5 related questions that can expand on this topic:

1. How do we handle equations with variables in the denominator, like in problem 1?
2. What is the difference between an identity and a contradiction?
3. How do you check if an equation has no solution or infinitely many solutions?
4. Can you explain the steps of distribution more thoroughly, like in problem 2?
5. How would you solve a quadratic equation if it appeared in any of these problems?

Tip: Always simplify both sides of an equation as much as possible to make solving easier.