We are given a linear sequence, and the slope (or common difference) of the function is 5. The sequence provided is:

$2, \, \_\_\_, \, 7, \, 12, \, 17, \, 22$

Since the slope is 5, this means the difference between consecutive terms of the sequence is 5.

Let's solve for the missing number and the first term in the sequence:

1. From 7 to 12, the difference is 5.
2. From 12 to 17, the difference is 5.
3. From 17 to 22, the difference is 5.

Now, we need to move backward to find the missing term and the first number. Since the common difference is 5, we subtract 5 from 7 to get the missing term:

$7 - 5 = 2$

So, the missing term is 2, meaning the sequence starts with 2, 2, 7, 12, 17, 22.

Now, let's move further backward to find the number before 2:

$2 - 5 = -3$

Therefore, the first number in the sequence should be -3.

The correct answer is A: -3.

Would you like more details on this solution or have any other questions?

Here are five follow-up questions related to this topic:

1. How would the sequence change if the slope were 7 instead of 5?
2. What is the general formula for a linear sequence?
3. How can we determine the nth term of a linear sequence?
4. If the first term was given as 5, how would you calculate the rest of the terms with the same slope?
5. What would happen if the sequence had a decreasing slope?

Tip: In arithmetic sequences, the common difference can help you move forward or backward through the sequence by adding or subtracting that value.