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We are given the triangle \( ZVW \), where:

- \( ZY \cong XW \)
- \( ZX = 5x + 17 \)
- \( YW = 10 - 2x \)
- \( YX = 3 \)

From the problem, since \( ZY \cong XW \), this means \( ZY = XW \). Also, we know that \( ZX = ZY + YX \) and \( YW = XW + YX \).

### Step 1: Express \( ZY \) and \( XW \) in terms of \( x \)
Since \( ZY \cong XW \), let both equal to some variable, \( ZY = XW = a \).

From the information:
- \( ZX = ZY + YX = a + 3 \)
- \( YW = XW + YX = a + 3 \)

### Step 2: Solve for \( x \)

We are given the equations:
- \( ZX = 5x + 17 \)
- \( YW = 10 - 2x \)

Set the expressions for \( ZX \) and \( YW \) equal to the expressions for \( a + 3 \):
\[
a + 3 = 5x + 17 \tag{1}
\]
\[
a + 3 = 10 - 2x \tag{2}
\]

### Step 3: Solve the system of equations

From equation (1):
\[
a = 5x + 17 - 3
\]
\[
a = 5x + 14 \tag{3}
\]

From equation (2):
\[
a = 10 - 2x - 3
\]
\[
a = 7 - 2x \tag{4}
\]

### Step 4: Set the two expressions for \( a \) equal to each other:
\[
5x + 14 = 7 - 2x
\]
Solve for \( x \):
\[
5x + 2x = 7 - 14
\]
\[
7x = -7
\]
\[
x = -1
\]

### Step 5: Find \( ZY = XW \)

Now, substitute \( x = -1 \) into either equation (3) or (4). Using equation (3):
\[
a = 5(-1) + 14
\]
\[
a = -5 + 14 = 9
\]

Thus, \( ZY = XW = 9 \).

### Final Answer:
\[
ZY = XW = 9
\]

Would you like further clarification on any part of this solution?

### Related Questions:
1. What kind of triangle is \( ZVW \) based on its side lengths?
2. Can you calculate the perimeter of triangle \( ZVW \)?
3. How does the property \( ZY \cong XW \) influence the symmetry of the triangle?
4. If \( x = -2 \), how would the side lengths change?
5. How does the triangle change if the given expressions for \( ZX \) and \( YW \) were different?

**Tip:** When solving systems of equations, always check if you can substitute variables to simplify calculations efficiently.

In the figure, ZY ≅ XW, ZX = 5x + 17, YW = 10 - 2x, and YX = 3. Find ZY and XW.

Learn how to solve for side lengths in a triangle where ZY ≅ XW, ZX = 5x + 17, and YW = 10 - 2x. This problem uses algebraic equations and geometric principles to find the values of ZY and XW. Suitable for students in Grades 9-10.

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