To find the value of $a$, let's first set up the equation based on the given information.

The arithmetic mean of $a + 7$ and $a + 3$ is calculated as follows:

$\text{Arithmetic Mean} = \frac{(a + 7) + (a + 3)}{2}$

Simplifying this:

$\text{Arithmetic Mean} = \frac{2a + 10}{2} = a + 5$

We are given that this mean is equal to $3a + 9$:

$a + 5 = 3a + 9$

Now, let's solve for $a$:

1. Subtract $a$ from both sides:

$5 = 2a + 9$

2. Subtract 9 from both sides:

$-4 = 2a$

3. Divide both sides by 2:

$a = -2$

Thus, the value of $a$ is $\boxed{-2}$.

Would you like further details, or do you have any other questions?

Here are some related questions for further practice:

1. What is the arithmetic mean of $2a + 4$ and $3a + 8$?
2. Solve for $a$ if the arithmetic mean of $4a + 2$ and $6a + 10$ equals $5a + 12$.
3. What is the geometric mean of $a + 2$ and $a + 6$?
4. If the arithmetic mean of $a$ and $b$ is 12, what is the sum of $a$ and $b$?
5. Find $a$ if the arithmetic mean of $a + 5$ and $a + 10$ equals $2a + 8$.
6. Determine the value of $a$ when the harmonic mean of $a + 2$ and $a + 4$ is $\frac{2a + 6}{3}$.
7. Solve for $b$ if the arithmetic mean of $b + 3$ and $b + 7$ is $5b + 2$.
8. Calculate the arithmetic mean of $a^2 + 1$ and $2a^2 - 3$.

Tip: When solving equations involving means, always start by simplifying the mean expression before setting it equal to the given value.