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We are given that triangles \( \triangle CSE \sim \triangle YJE \), meaning they are similar. This allows us to set up a proportion between the corresponding sides of the two triangles to solve for \( EY \).

Let's identify the corresponding sides based on the similarity:

- \( \frac{CS}{YJ} = \frac{SE}{JE} = \frac{CE}{YE} \)

### Step-by-Step Solution:

1. **Identify Known Values and Expressions:**
   - \( CS = 18 \)
   - \( SE = x + 1 \)
   - \( YJ = 10 \)
   - \( JE = x - 2 \)
   
2. **Set Up a Proportion:**
   Since \( \triangle CSE \sim \triangle YJE \), the ratios of corresponding sides are equal. We can use:
   \[
   \frac{CS}{YJ} = \frac{SE}{JE}
   \]
   
   Plugging in the values:
   \[
   \frac{18}{10} = \frac{x + 1}{x - 2}
   \]

3. **Solve for \( x \):**
   Cross-multiply to clear the fraction:
   \[
   18(x - 2) = 10(x + 1)
   \]
   Expanding both sides:
   \[
   18x - 36 = 10x + 10
   \]
   Rearrange terms to isolate \( x \):
   \[
   18x - 10x = 36 + 10
   \]
   \[
   8x = 46
   \]
   \[
   x = \frac{46}{8} = 5.75
   \]

4. **Calculate \( EY \):**
   Substitute \( x = 5.75 \) into \( EY = x + 1 \):
   \[
   EY = 5.75 + 1 = 6.75
   \]

### Answer:
\[
EY = 6.75
\]

Would you like more details or have any questions?

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#### Related Questions:
1. How do you determine corresponding sides in similar triangles?
2. What other ratios could we use if different values were given?
3. How do you handle proportions when variables appear on both sides of the fraction?
4. Why is it important that the triangles are similar?
5. Can you solve for \( JE \) if given \( EY \)?

#### Tip:
When dealing with similar triangles, always double-check the order of corresponding sides to avoid mistakes in setting up proportions.

Given that triangles ΔCSE ~ ΔYJE, find EY.

Learn how to find the length of EY given that triangles ΔCSE and ΔYJE are similar. This solution involves setting up and solving proportions using corresponding sides of similar triangles. Suitable for students in Grades 9-10.

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