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The image shows a rhombus with angles and side lengths represented by algebraic expressions. Here's a breakdown of how to solve for \( x \) and \( y \):

### 1. Identifying Key Information:
- **JK = KL = LM = MJ**: Since this is a rhombus, all sides are congruent. So, the expressions for side lengths are equal.
  - \( JK = 4x + 17 \)
  - \( LM = 8x - 3 \)
  Therefore, \( 4x + 17 = 8x - 3 \).

- **Opposite Angles of a Rhombus** are congruent. So:
  - \( \angle J = \angle L \)
  - \( \angle K = \angle M = 108^\circ \)
  
  We are given that:
  - \( \angle J = (7y - 5)^\circ \).
  
  The sum of the angles in a rhombus is \( 360^\circ \). So, using the fact that \( \angle K = \angle M = 108^\circ \), we can solve for \( y \) from:
  \[
  2(7y - 5) + 2(108) = 360.
  \]

### 2. Solving the Equations:

#### Equation for \( x \):
\[
4x + 17 = 8x - 3
\]
Solve for \( x \):
\[
17 + 3 = 8x - 4x
\]
\[
20 = 4x
\]
\[
x = 5
\]

#### Equation for \( y \):
\[
2(7y - 5) + 2(108) = 360
\]
Simplify:
\[
14y - 10 + 216 = 360
\]
\[
14y + 206 = 360
\]
\[
14y = 360 - 206
\]
\[
14y = 154
\]
\[
y = 11
\]

### Final Answers:
- \( x = 5 \)
- \( y = 11 \)

Would you like further details or explanations on any part of the solution? Here are some related questions:

1. Why are opposite angles in a rhombus congruent?
2. How do you derive the equation for the sum of the angles in a quadrilateral?
3. What are the properties of a rhombus that make the sides equal?
4. Can this method be used for other types of quadrilaterals like squares or rectangles?
5. How would you solve the equations if given different angle measurements?

**Tip:** In a rhombus, diagonals bisect the angles, which can help in more complex problems involving diagonals and angles!

Solve for x and y in the rhombus with side lengths and angles expressed algebraically.

Learn how to solve a geometry problem involving a rhombus where sides and angles are represented with algebraic expressions. This step-by-step solution includes solving linear equations and applying the properties of rhombus geometry. Suitable for students in Grades 8-10.

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