Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. A charitable organization in Castroville is hosting a black tie benefit. Yesterday, the organization sold 92 regular tickets and 88 VIP tickets, raising $20,828. Today, 53 regular tickets and 87 VIP tickets were sold, bringing in a total of $18,352. How much do the different ticket types cost? A regular ticket costs $_____, and a VIP ticket costs $_____.

Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.  A charitable organization in Castroville is hosting a black tie benefit. Yesterday, the organization sold 92 regular tickets and  88 VIP tickets, raising $20,828. Today, 53 regular tickets and  87 VIP tickets were sold, bringing in a total of $18,352. How much do the different ticket types cost?  A regular ticket costs $_____, and a VIP ticket costs $_____.

Bytelearn · Accepted Answer

Denote Ticket Costs: Let's denote the cost of a regular ticket as $ r $ and the cost of a VIP ticket as $ v $. We are given that 92 regular tickets and 88 VIP tickets were sold for a total of $20,828. This can be represented by the equation: $$ 92r + 88v = 20828 $$Equations Given: We are also given that 53 regular tickets and 87 VIP tickets were sold for a total of $18,352. This can be represented by the equation: $$ 53r + 87v = 18352 $$Solving System of Equations: We now have a system of two equations with two variables: $$ 92r + 88v = 20828 $$ $$ 53r + 87v = 18352 $$ We can solve this system using either substitution or elimination. Let's use the elimination method to solve for one of the variables.Elimination Method: To eliminate one of the variables, we can multiply the second equation by 92 and the first equation by 53, so that the coefficients of $ r $ will be the same in both equations: $$ 92 \times (53r + 87v) = 92 \times 18352 $$ $$ 53 \times (92r + 88v) = 53 \times 20828 $$Multiplying Equations: After multiplying, we get the new system of equations: $$ 4876r + 8084v = 1687584 $$ $$ 4876r + 4664v = 1103896 $$ Now we can subtract the second equation from the first to eliminate $ r $: $$ (4876r + 8084v) - (4876r + 4664v) = 1687584 - 1103896 $$Subtracting Equations: Subtracting the equations gives us: $$ 4876r - 4876r + 8084v - 4664v = 1687584 - 1103896 $$ $$ 3420v = 583688 $$ Now we can solve for $ v $ by dividing both sides by 3420: $$ v = \frac{583688}{3420} $$ $$ v = 170.7 $$Solving for v: Now that we have the value for $ v $, we can substitute it back into one of the original equations to solve for $ r $. Let's use the first equation: $$ 92r + 88v = 20828 $$ $$ 92r + 88 \times 170.7 = 20828 $$ $$ 92r + 15021.6 = 20828 $$ $$ 92r = 20828 - 15021.6 $$ $$ 92r = 5806.4 $$ $$ r = \frac{5806.4}{92} $$ $$ r = 63.11 $$

Full solution

More problems from Solve a system of equations using any method: word problems

Related topics