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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.

Point M is in between A and B on line segment AB. Find AM, if AM = 4x + 7, MB = 5x - 8, and AB = 53.

Learn how to solve for the length of segment AM in a line segment with point M between A and B. The problem involves using linear equations and the segment addition postulate, where AM = 4x + 7, MB = 5x - 8, and AB = 53. Suitable for Grades 8-10.

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