1. **State the problem:** Eric and Lucia play a game where points are earned by traveling to asteroids and planets. Eric travels to 1 asteroid and 8 planets, earning 132 points. Lucia travels to 3 asteroids and 7 planets, earning 141 points. We need to find the points earned for traveling to one asteroid and one planet. 2. **Set variables:** Let $x$ be the points for one asteroid and $y$ be the points for one planet. 3. **Write equations from the problem:** $$\begin{cases} 1x + 8y = 132 \\ 3x + 7y = 141 \end{cases}$$ 4. **Solve the system of equations:** Multiply the first equation by 3: $$3x + 24y = 396$$ Subtract the second equation from this: $$\cancel{3x} + 24y - (\cancel{3x} + 7y) = 396 - 141$$ $$24y - 7y = 255$$ $$17y = 255$$ 5. **Find $y$:** $$y = \frac{255}{17} = 15$$ 6. **Substitute $y=15$ into the first equation:** $$x + 8(15) = 132$$ $$x + 120 = 132$$ 7. **Find $x$:** $$x = 132 - 120 = 12$$ **Final answer:** Players earn $12$ points for traveling to an asteroid and $15$ points for traveling to a planet.