A charitable organization in Lowell is hosting a black tie benefit. Yesterday, the organization sold 30 regular tickets and 58 VIP tickets, raising $9,346. Today, 26 regular tickets and 59 VIP tickets were sold, bringing in a total of $9,209. How much do the regular and VIP ticket types cost?
Let R be the cost of a regular ticket and V be the cost of a VIP ticket.
Yesterday, the organization sold 30 regular tickets, so they made 30R.
They also sold 58 VIP tickets, so they made 58V.
Yesterday's tickets brought in $9,346, so we know that 30R + 58V = 9346. Equation 1.
Today, the organization sold 26 regular tickets, so they made 26R.
They also sold 59 VIP tickets, so they made 59V.
Today's tickets brought in $9,209, so we know that 26R + 59V = 9209. Equation 2.
We can solve this system of equations by multiplying Equation 2 by 2.
52R + 118V = 18418. Equation 3.
We then subtract Equation 1 from Equation 3.
52R + 118V - (30R + 58V) = 18418 - 9346.
22R + 60V = 9072.
We can divide this equation by 2 to simplify it further.
11R + 30V = 4536. Equation 4.
We can multiply Equation 4 by 58 and subtract it from multiplied Equation 1 to get:
58(30R + 58V) - (11R + 30V) = 58(9346) - 4536.
1740R + 3364V - 11R - 30V = 541548 - 4536,
1729R + 3334V = 537012.
We subtract Equation 2 multiplied by 5 from this equation
(1729R + 3334V) - 5(26R + 59V) = 537012 - 5(9209),
1629R + 3089V = 483753.
We can then solve this system of equations subtracting Equation 5 from Equation 4.
(11R + 30V) - (1629R + 3089V) = 4536 - 483753,
-1618R - 3059V = -479217.
We can multiply Equation 4 by -3059 to simplify it further.
-3059(11R + 30V) = -3059(4536),
-33649R - 91770V = -13878484,
we can add this to Equation 6 to get:
(-1618R - 3059V) + (-33649R - 91770V) = -479217 - 13878484,
-35267R - 94829V = -14357655.
Now, let's solve this system of equations.
-1618R - 3059V + -33649R - 91770V = -479217 - 13878484,
-35267R - 94829V = -14357655,
Multiplying Equation 7 by -1 to leave R positive.
1618R + 3059V + 33649R + 91770V = 479217 + 13878484,
35267R + 94829V = 14357655.
Adding Equation 7 and Equation 8 together
-35267R - 94829V + 35267R + 94829V = -14357655 + 14357655,
0 = 0.
Since 0 = 0, any value of R and V satisfies this original system of equations.
Therefore, there are infinitely many solutions for R and V.