To solve the problem, let's use the information provided:

Given:

• $AB = 7x + 656$
• $BC = 7x + 627$
• $CD = 8x + 760$
• $AD = 107$

Since points $A$, $B$, $C$, and $D$ are collinear and in that order, the total length $AD$ is the sum of $AB$, $BC$, and $CD$:

$AD = AB + BC + CD$

Substituting the given expressions:

$107 = (7x + 656) + (7x + 627) + (8x + 760)$

Now, combine like terms:

$107 = 7x + 656 + 7x + 627 + 8x + 760$

Simplify the equation:

$107 = 22x + 2043$

Next, solve for $x$:

$22x = 107 - 2043$

$22x = -1936$

$x = \frac{-1936}{22} = -88$

Now that we have $x = -88$, substitute it back into the expression for $AB$:

$AB = 7(-88) + 656 = -616 + 656 = 40$

So, the length of $AB$ is 40 units.

Would you like any further explanation or details?

Here are five related questions to consider:

1. How would the problem change if $AB$ was given instead of $AD$?
2. Can you determine the lengths of $BC$ and $CD$ using the value of $x$?
3. What happens if the value of $x$ was positive? How would it affect the lengths?
4. How would the solution change if the points were not collinear?
5. What if one of the segments was instead defined by a quadratic expression?

Tip: When working with equations, always remember to combine like terms and simplify as much as possible before solving for the unknowns.