1. **State the problem:** We have a kite-shaped quadrilateral with sides labeled as follows: top side = $3y$, left side = $5x - 15$, right side = $2x + 3$, and bottom side = $6y - 2$. The perimeter is given as 86 ft. 2. **Recall the property of a kite:** A kite has two pairs of adjacent sides equal. So, the left side equals the top side or bottom side, and the right side equals the other adjacent side. 3. **Set up equations based on the kite property:** - Left side equals right side: $5x - 15 = 2x + 3$ - Top side equals bottom side: $3y = 6y - 2$ 4. **Solve for $x$:** $$5x - 15 = 2x + 3$$ $$5x - 2x = 3 + 15$$ $$3x = 18$$ $$x = \frac{18}{3} = 6$$ 5. **Solve for $y$:** $$3y = 6y - 2$$ $$3y - 6y = -2$$ $$-3y = -2$$ $$y = \frac{-2}{-3} = \frac{2}{3}$$ 6. **Verify the perimeter:** Perimeter $P = 3y + (5x - 15) + (2x + 3) + (6y - 2)$ Substitute $x=6$, $y=\frac{2}{3}$: $$P = 3 \times \frac{2}{3} + (5 \times 6 - 15) + (2 \times 6 + 3) + (6 \times \frac{2}{3} - 2)$$ $$= 2 + (30 - 15) + (12 + 3) + (4 - 2)$$ $$= 2 + 15 + 15 + 2 = 34$$ Since the sum is 34, but the problem states the perimeter is 86, the assumption that left equals right and top equals bottom is incorrect. 7. **Try the other pairings:** Left side equals bottom side: $5x - 15 = 6y - 2$ Right side equals top side: $2x + 3 = 3y$ 8. **Express $y$ from the second equation:** $$3y = 2x + 3$$ $$y = \frac{2x + 3}{3}$$ 9. **Substitute $y$ into the first equation:** $$5x - 15 = 6 \times \frac{2x + 3}{3} - 2$$ $$5x - 15 = 2(2x + 3) - 2$$ $$5x - 15 = 4x + 6 - 2$$ $$5x - 15 = 4x + 4$$ $$5x - 4x = 4 + 15$$ $$x = 19$$ 10. **Find $y$:** $$y = \frac{2(19) + 3}{3} = \frac{38 + 3}{3} = \frac{41}{3}$$ 11. **Calculate the perimeter with these values:** $$P = 3y + (5x - 15) + (2x + 3) + (6y - 2)$$ $$= 3 \times \frac{41}{3} + (5 \times 19 - 15) + (2 \times 19 + 3) + (6 \times \frac{41}{3} - 2)$$ $$= 41 + (95 - 15) + (38 + 3) + (82 - 2)$$ $$= 41 + 80 + 41 + 80 = 242$$ This is too large, so the perimeter does not match 86. 12. **Use the perimeter equation directly:** $$3y + (5x - 15) + (2x + 3) + (6y - 2) = 86$$ Simplify: $$3y + 5x - 15 + 2x + 3 + 6y - 2 = 86$$ $$5x + 2x + 3y + 6y - 15 + 3 - 2 = 86$$ $$7x + 9y - 14 = 86$$ $$7x + 9y = 100$$ 13. **Use the kite property: adjacent sides equal. Try left = right:** $$5x - 15 = 2x + 3$$ $$3x = 18$$ $$x = 6$$ 14. **Substitute $x=6$ into perimeter equation:** $$7(6) + 9y = 100$$ $$42 + 9y = 100$$ $$9y = 58$$ $$y = \frac{58}{9}$$ 15. **Final answers:** $$x = 6, \quad y = \frac{58}{9}$$