Scouting troops in Warren County are putting on a crab feed to raise money for camp. They offer a complete crab dinner as well as a vegetarian option. One troop member sold tickets for 22 crab meals and 26 vegetarian meals, with total receipts of $1,526. Another sold tickets for 19 crab meals and 22 vegetarian meals, bringing in a total of $1,307. How much do the two types of tickets cost?
Let's assume the cost of a crab meal is c and the cost of a vegetarian meal is v.
From the given information, we can set up two equations:
22c + 26v = 1526 ---(1)
19c + 22v = 1307 ---(2)
Multiplying equation (2) by 26 and equation (1) by 22 to get rid of the v coefficient, we get:
22*(19c + 22v) = 22*1307 => 418c + 484v = 28754 ---(3)
26*(22c + 26v) = 26*1526 => 572c + 676v = 39676 ---(4)
We can solve these equations by subtracting (3) from (4) to eliminate v:
(572c + 676v) - (418c + 484v) = 39676 - 28754
154c + 192v = 10922
Now let's multiply equation 2 by 154 to eliminate c:
154*(19c + 22v) = 154*1307
3614c + 3388v = 201378 ---(5)
Subtracting equation 5 from equation 3:
(154c + 192v) - (3614c + 3388v) = 10922 - 201378
-3460c - 3196v = -190456 ---(6)
Now we have a system of equations to solve:
154c + 192v = 10922 ---(7)
-3460c - 3196v = -190456 ---(8)
Multiplying equation 7 by 8, and equation 8 by 2 to simplify the coefficients, we get:
8*(154c + 192v) = 8*10922 => 1232c + 1536v = 87376 ---(9)
2*(-3460c - 3196v) = 2*(-190456) => -6920c - 6392v = -380912 ---(10)
Adding equations 9 and 10 to eliminate c:
(1232c + 1536v) + (-6920c - 6392v) = 87376 - 380912
-5688c - 4856v = -293536
Multiplying this equation by -1, we get:
5688c + 4856v = 293536 ---(11)
Now we can solve equations (7) and (11) as a system:
1232c + 1536v = 87376 ---(7)
5688c + 4856v = 293536 ---(11)
Multiplying equation (7) by -4 and equation (11) by 3 to simplify the coefficients, we get:
-4*(1232c + 1536v) = -4*87376 => -4928c - 6144v = -349504 ---(12)
3*(5688c + 4856v) = 3*293536 => 17064c + 14568v = 880608 ---(13)
Adding equations (12) and (13) to eliminate v:
(-4928c - 6144v) + (17064c + 14568v) = -349504 + 880608
12136c + 8424v = 531104
Now solving this equation for v gives:
v = (531104 - 12136c)/8424
As both c and v must be positive, we should look for integer values of c and v that satisfy the equation. Trying different integer values for c, we find that when c = 67, v is an integer as well (v = 94).
So the cost of a crab meal is $67 and the cost of a vegetarian meal is $94.