https://file.aiquickdraw.com/mathbot/file/17255170871336l7r833l.jpg

The image shows a trapezoid with two expressions for the lengths of the non-parallel sides. The left side is labeled \( 4(y - 3) \) and the right side is labeled \( 2y + 2 \).

If you are solving for \( y \), assuming that the two sides are congruent (which is a reasonable assumption in the case of an isosceles trapezoid), we can set the expressions for the lengths of the sides equal to each other:

\[
4(y - 3) = 2y + 2
\]

Let's solve for \( y \).

### Step 1: Expand both sides
\[
4(y - 3) = 4y - 12
\]
Thus, the equation becomes:
\[
4y - 12 = 2y + 2
\]

### Step 2: Move all terms involving \( y \) to one side
\[
4y - 2y = 2 + 12
\]
\[
2y = 14
\]

### Step 3: Solve for \( y \)
\[
y = \frac{14}{2} = 7
\]

Thus, \( y = 7 \).

If you'd like any further explanations or details, feel free to ask!

Here are some related questions:
1. How do we calculate the perimeter of this trapezoid?
2. How can we find the area of a trapezoid given the height and base lengths?
3. What conditions are necessary for a trapezoid to be isosceles?
4. What happens to the lengths of the sides if \( y \) is altered?
5. Can this problem be extended to solve for the bases of the trapezoid?

**Tip**: Always check if sides are congruent before setting up equations in geometric problems!

Solve for y in the trapezoid where one side is 4(y - 3) and the other is 2y + 2.

This problem involves solving for y in a trapezoid with congruent sides given by the expressions 4(y - 3) and 2y + 2. It covers algebraic concepts like solving linear equations and geometric properties of trapezoids.

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