The problem states:

• Point $M$ is between points $A$ and $B$ on line segment $\overline{AB}$.
• $AM = 4x + 7$, $MB = 5x - 8$, and $AB = 53$.

We are asked to find the length of $AM$.

Step-by-step solution:

Since point $M$ is between $A$ and $B$, we know the total length of $\overline{AB}$ is the sum of the lengths of $AM$ and $MB$: $AM + MB = AB$ Substitute the given expressions for $AM$, $MB$, and $AB$: $(4x + 7) + (5x - 8) = 53$ Now, simplify the left-hand side: $4x + 5x + 7 - 8 = 53$ $9x - 1 = 53$ Add 1 to both sides: $9x = 54$ Solve for $x$: $x = \frac{54}{9} = 6$

Finding $AM$:

Now that we have $x = 6$, substitute it back into the expression for $AM$: $AM = 4x + 7 = 4(6) + 7 = 24 + 7 = 31$

Thus, the length of $AM$ is 31 units.

Would you like further details or have any questions? Here are some related questions:

1. How would the result change if the total length $AB$ were different?
2. What is the length of $MB$ with the value of $x = 6$?
3. Can this method be generalized to solve other problems with similar equations?
4. How can you verify if the value of $AM$ is correct based on the total length of $AB$?
5. What would happen if $M$ were not between $A$ and $B$?

Tip: Always check the total length by adding $AM$ and $MB$ to verify the solution.