There are only two unknowns -- the number of pies and the number of cakes. With two unknowns you only need two equations or inequalities.
The statement of the problem doesn't make it clear whether we are to use equations or inequalities; so let's try equations first.
Let p = number of pies
Let c = number of cakes
The total cost of the desserts, at $10 per pie and $20 per cake, must be $200:
The number of people to be served, at 12 per pie and 24 per cake, is 210:
The first equation is equivalent to ; the second equation is equivalent to
Obviously there is no common solution to those two equations.
So apparently we are to use inequalities -- the total cost must be AT MOST $200; and the number of people to be fed must be AT LEAST 210.
Then the inequalities are equivalent to and .
And that pair of inequalities is equivalent to the compound inequality
We can assume that the numbers of pies and cakes must be whole numbers. So any pair of whole numbers c and p that satisfy that inequality are the possible numbers of pies and cakes.
There are numerous such pairs; here are a few of them....
pies cakes cost people served
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0 9 0(10)+ 9(20) = 180 0(12)+ 9(24) = 216
0 10 0(10)+10(20) = 200 0(12)+10(24) = 240
1 9 1(10)+ 9(20) = 190 1(12)+ 9(24) = 228
2 8 2(10)+ 8(20) = 180 2(12)+ 8(24) = 216
2 9 2(10)+ 9(20) = 200 2(12)+10(24) = 264
...
18 0 18(10)+ 0(20) = 180 18(12)+ 0(24) = 216
18 1 18(10)+ 1(20) = 200 18(12)+ 1(24) = 240
...