https://file.aiquickdraw.com/mathbot/file/1726078434344yokaqztn.jpg

The image shows two intersecting lines with angles formed by a transversal crossing two parallel lines. The two angles are represented as:

- \( (3y + 53)^\circ \)
- \( (7y - 55)^\circ \)

Since these two angles are alternate interior angles (based on the diagram with the transversal crossing parallel lines), they are equal. Therefore, we can set up the equation:

\[
3y + 53 = 7y - 55
\]

Now, let’s solve for \( y \).

\[
3y + 53 = 7y - 55
\]
Subtract \( 3y \) from both sides:

\[
53 = 4y - 55
\]
Add \( 55 \) to both sides:

\[
108 = 4y
\]
Now, divide by \( 4 \):

\[
y = 27
\]

Thus, the value of \( y \) is \( \boxed{27} \).

Would you like a further breakdown, or have any questions?

### Related Questions:
1. What is the significance of alternate interior angles in parallel lines?
2. How do corresponding angles differ from alternate interior angles?
3. Can angles like these be supplementary or complementary, and how?
4. How do we prove that two lines are parallel using angle relationships?
5. What other geometric properties can be derived from this type of diagram?

**Tip:** In geometry problems involving parallel lines and a transversal, alternate interior angles are always congruent (equal in measure).

This problem demonstrates solving for y using the concept of alternate interior angles formed by a transversal crossing parallel lines. It involves algebraic manipulation and understanding geometric angle relationships. Suitable for students in Grades 7-9.

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