At a county fair, adults' tickets sold for $\$5.50$, senior citizens' tickets for $\$4.00$, and children's tickets for $\$1.50$. On the opening day, the number of children's and senior's tickets sold was 30 more than half the number of adults' tickets sold. The number of senior citizens' tickets sold was $5$ more than four times the number of children's tickets. How many of each type of ticket were sold if the total receipts from the ticket sales were $\$14,970$?
This is how I solved it, but I did something wrong and don't know what. $$c+s=\frac 12a+30$$ $$s=4c+5$$ $$5.5a+4s+1.5c=14970$$ $$5.5a+16c+20+1.5c=14970$$ $$5.5a+1.75a+87.5+20=14970$$ $$7.25a=1486.25$$ $$a=205$$ $$c=25.5, s=107$$
$$c+s=\frac 12a+30$$ $$s=4c+5$$ $$5.5a+4s+1.5c=14970$$ $$5.5a+16c+20+1.5c=14970$$$$c=\frac 1{10}a+5$$ $$5.5a+1.75a+87.5+20=14970$$ $$\text{ *** mistake here -> }7.25a=14862.5$$ $$a=2050$$ $$c=210, s=845$$