https://file.aiquickdraw.com/mathbot/file/17267868091429gubifr1.jpg

In the diagram, we have the following information:

- Line \( y \) bisects \( \overline{AC} \) at point \( B \), meaning that the lengths of \( AB \) and \( BC \) are equal.
- The length of \( AB \) is given as \( 4 - 5x \), and the length of \( BC \) is \( 2x + 25 \).

Since line \( y \) bisects \( \overline{AC} \), we know that:
\[
AB = BC
\]

Thus, we can set the two expressions equal to each other:
\[
4 - 5x = 2x + 25
\]

Now, let's solve for \( x \).

1. Combine like terms:
\[
4 - 5x = 2x + 25
\]
\[
4 - 25 = 2x + 5x
\]
\[
-21 = 7x
\]

2. Solve for \( x \):
\[
x = \frac{-21}{7} = -3
\]

Now that we know \( x = -3 \), we can substitute it back into the expressions for \( AB \) and \( BC \) to verify the lengths are equal.

For \( AB \):
\[
AB = 4 - 5(-3) = 4 + 15 = 19
\]

For \( BC \):
\[
BC = 2(-3) + 25 = -6 + 25 = 19
\]

Thus, both \( AB \) and \( BC \) are 19 units long, which confirms that the line \( y \) bisects \( \overline{AC} \) as required.

### Summary:
- \( x = -3 \)
- \( AB = BC = 19 \)

Would you like more details on any part of the solution? 

Here are 5 related questions:
1. How would the solution change if line \( y \) did not bisect \( \overline{AC} \)?
2. What is the length of \( AC \) after solving for \( x \)?
3. Can you find the coordinates of points \( A \) and \( C \) if the bisection point \( B \) is known?
4. How do we verify that the midpoint theorem holds in this case?
5. What geometric properties are preserved by the bisection of a line?

**Tip**: When setting up equations from a geometric diagram, always focus on equal segments or angles to form relationships.

If line y bisects AC, AB = 4 - 5x and BC = 2x + 25. Solve for x and find the lengths of AB and BC.

Learn how to solve the geometric problem where line y bisects AC and the lengths of AB and BC are expressed algebraically. This problem covers key concepts in algebra and geometry, suitable for students in Grades 7-9.

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