1. **Problem statement:** A football team plays 30 matches. Each win gives 5 points, each draw gives 2 points, and each loss deducts 1 point. The team scored a total of 108 points. We need to find the largest possible number of wins. 2. **Define variables:** Let $w$ = number of wins, $d$ = number of draws, and $l$ = number of losses. 3. **Equations:** - Total matches: $$w + d + l = 30$$ - Total points: $$5w + 2d - l = 108$$ 4. **Express $l$ from the first equation:** $$l = 30 - w - d$$ 5. **Substitute $l$ into the points equation:** $$5w + 2d - (30 - w - d) = 108$$ 6. **Simplify:** $$5w + 2d - 30 + w + d = 108$$ $$6w + 3d - 30 = 108$$ $$6w + 3d = 138$$ 7. **Divide entire equation by 3:** $$2w + d = 46$$ 8. **Express $d$:** $$d = 46 - 2w$$ 9. **Recall $l = 30 - w - d$:** $$l = 30 - w - (46 - 2w) = 30 - w - 46 + 2w = w - 16$$ 10. **Constraints:** - $w, d, l$ must be non-negative integers. - So, $d = 46 - 2w \\geq 0 \implies 2w \leq 46 \implies w \leq 23$ - $l = w - 16 \\geq 0 \implies w \geq 16$ 11. **Combine constraints:** $$16 \leq w \leq 23$$ 12. **Maximize $w$:** The largest integer $w$ in this range is 23. 13. **Check $d$ and $l$ for $w=23$:** $$d = 46 - 2(23) = 46 - 46 = 0$$ $$l = 23 - 16 = 7$$ 14. **Verify total matches:** $$23 + 0 + 7 = 30$$ 15. **Verify total points:** $$5(23) + 2(0) - 7 = 115 - 7 = 108$$ **Final answer:** The largest possible number of wins is **23**.