Mr frankel bought 7tickets to puppet show and spent $43. he bought a combination of child tickets and adult tickets for $9.what system of eq

Mr frankel bought 7tickets to puppet show and spent $43. he bought a combination of child tickets and adult tickets for $9.what system of equation determine the number of adult tickets, a, and the number of child tickets,c, he bought?

I suspect that the question was supposed to include the cost of the child's ticket but I will play with it anyway. 

Let the cost of the child ticket be R (R for reduced price)

9a+Rc=43

a+c=7

We know that a and c have to be whole numbers. And that all the pronumerals have positive values.

R must be exact to 2 decimal places (cents)

multiples of 9 are 9,18,27,36  ..  So there must be 1,2,3,or 4 adults

If there was 1 adult then there are 6 children and their tickets cost 43-9=$34    34/6=5.66666 no not exact!

If there was 2 adult then there are 5 children and their tickets cost 43-18=$25    25/5=$5 Sounds good

If there was 3 adult then there are 4 children and their tickets cost 43-27=$16    16/4=$4 sounds good

If there was 4 adult then there are 3 children and their tickets cost 43-36=$7    7/3=$2.333 no not exact!

----------------------------------------------------------------------------

So he may have bought

2adult tickets for $9each and 5 child tickets for $5 each  OR

3adult tickets for $9each and 4 child tickets for $4 each