To solve this, let's carefully analyze the information given:

1. $e$ can be written as: $e = \frac{a + c}{d}$

2. The value of $a$ is: $a = \frac{b + c}{d}$

3. The value of $b$ is: $b = \frac{a + c}{d}$

We need to find the value of $e$. To do that, we can use substitution. Notice that from the given equations, both $e$ and $b$ are expressed as $\frac{a + c}{d}$. This means: $e = b$

Now let's substitute the expression for $b$ from the third equation: $b = \frac{a + c}{d}$ We have: $e = \frac{a + c}{d}$

Since $e$ and $a$ are related, we substitute the expression for $a$ from the second equation: $a = \frac{b + c}{d}$ Now, using the fact that $b = e$, we substitute again: $a = \frac{e + c}{d}$

We already have $e = \frac{a + c}{d}$, so let's substitute the value of $a$ we found: $e = \frac{\frac{e + c}{d} + c}{d}$

To simplify, let's multiply both the numerator and the denominator by $d$ to eliminate the fractions: $e = \frac{e + c + cd}{d^2}$

Now, we rearrange the equation to isolate $e$: $e \cdot d^2 = e + c + cd$ $e \cdot d^2 - e = c + cd$ $e(d^2 - 1) = c(d + 1)$

Finally, we can express $e$ as: $e = \frac{c(d + 1)}{d^2 - 1}$

This is the simplified final value of $e$.

Would you like further details or clarification on any of these steps? Here are 5 related questions you might find helpful:

1. How do you solve similar equations by substitution?
2. What happens if $c$ or $d$ equals zero in this expression?
3. Can this process be used to solve for $a$ and $b$ directly as well?
4. How does the simplification of fractions work in multi-step problems?
5. What are common scenarios where circular dependencies like this one appear?

Tip: When faced with multiple equations where variables depend on each other, substitution and rearrangement are key strategies to isolate and solve for each variable.