1. **State the problem:** We have two similar triangles. The longest side of the first triangle is 8 less than 3 times the longest side of the second triangle. The shortest sides of the triangles are 15 and 6, respectively. We need to find the lengths of the longest sides of both triangles. 2. **Set variables:** Let $x$ be the longest side of the second triangle. Then the longest side of the first triangle is $3x - 8$. 3. **Use similarity ratio:** Since the triangles are similar, the ratio of corresponding sides is the same. So, $$\frac{15}{6} = \frac{3x - 8}{x}$$ 4. **Simplify the ratio on the left:** $$\frac{15}{6} = \frac{5}{2}$$ So the equation becomes: $$\frac{5}{2} = \frac{3x - 8}{x}$$ 5. **Cross multiply:** $$5 \times x = 2 \times (3x - 8)$$ $$5x = 6x - 16$$ 6. **Isolate $x$:** $$5x - 6x = -16$$ $$\cancel{5x} - \cancel{6x} = -16$$ $$-x = -16$$ 7. **Solve for $x$:** $$x = 16$$ 8. **Find the longest side of the first triangle:** $$3x - 8 = 3 \times 16 - 8 = 48 - 8 = 40$$ **Final answer:** The longest side of the second triangle is $16$, and the longest side of the first triangle is $40$.