Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ A charitable organization in Newport is hosting a black tie benefit. Yesterday, the organization sold $92$ regular tickets and $94$ VIP tickets, raising $\$21,408$ . Today, $92$ regular tickets and $93$ VIP tickets were sold, bringing in a total of $\$21,238$ . How much do the different ticket types cost? $\newline$ A regular ticket costs $\$$ _, and a VIP ticket costs $\$$ _.

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Grade 8

Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

A charitable organization in Newport is hosting a black tie benefit. Yesterday, the organization sold

92

regular tickets and

94

VIP tickets, raising

\$21,408

. Today,

92

regular tickets and

93

VIP tickets were sold, bringing in a total of

\$21,238

. How much do the different ticket types cost?

\newline

A regular ticket costs

\$

_____, and a VIP ticket costs

\$

_____.

Define variables: Let's define two variables: let $r$ be the price of a regular ticket, and $v$ be the price of a VIP ticket.
Write equations: We can write two equations based on the information given. For the first day, the equation is $92r + 94v = 21408$ . For the second day, the equation is $92r + 93v = 21238$ .
Eliminate variable: To use elimination, we need to eliminate one of the variables. We can subtract the second equation from the first to eliminate $r$ . $(92r + 94v) - (92r + 93v) = 21408 - 21238$ .
Substitute and solve: Performing the subtraction, we get $92r - 92r + 94v - 93v = 21408 - 21238$ , which simplifies to $v = 170$ .
Calculate final prices: Now that we have the value of $v$ , we can substitute it back into one of the original equations to find $r$ . Let's use the second day's equation: $92r + 93(170) = 21238$ .
Calculate final prices: Now that we have the value of $v$ , we can substitute it back into one of the original equations to find $r$ . Let's use the second day's equation: $92r + 93(170) = 21238$ . Substitute the value of $v$ into the equation: $92r + 15810 = 21238$ .
Calculate final prices: Now that we have the value of $v$ , we can substitute it back into one of the original equations to find $r$ . Let's use the second day's equation: $92r + 93(170) = 21238$ . Substitute the value of $v$ into the equation: $92r + 15810 = 21238$ . Now, we solve for $r$ : $92r = 21238 - 15810$ .
Calculate final prices: Now that we have the value of $v$ , we can substitute it back into one of the original equations to find $r$ . Let's use the second day's equation: $92r + 93(170) = 21238$ . Substitute the value of $v$ into the equation: $92r + 15810 = 21238$ . Now, we solve for $r$ : $92r = 21238 - 15810$ . Calculate the right side of the equation: $92r = 5428$ .
Calculate final prices: Now that we have the value of $v$ , we can substitute it back into one of the original equations to find $r$ . Let's use the second day's equation: $92r + 93(170) = 21238$ . Substitute the value of $v$ into the equation: $92r + 15810 = 21238$ . Now, we solve for $r$ : $92r = 21238 - 15810$ . Calculate the right side of the equation: $92r = 5428$ . Divide both sides by $92$ to find $r$ : $r$ $0$ .
Calculate final prices: Now that we have the value of $v$ , we can substitute it back into one of the original equations to find $r$ . Let's use the second day's equation: $92r + 93(170) = 21238$ . Substitute the value of $v$ into the equation: $92r + 15810 = 21238$ . Now, we solve for $r$ : $92r = 21238 - 15810$ . Calculate the right side of the equation: $92r = 5428$ . Divide both sides by $92$ to find $r$ : $r$ $0$ . Perform the division: $r$ $1$ .
Calculate final prices: Now that we have the value of $v$ , we can substitute it back into one of the original equations to find $r$ . Let's use the second day's equation: $92r + 93(170) = 21238$ . Substitute the value of $v$ into the equation: $92r + 15810 = 21238$ . Now, we solve for $r$ : $92r = 21238 - 15810$ . Calculate the right side of the equation: $92r = 5428$ . Divide both sides by $92$ to find $r$ : $r$ $0$ . Perform the division: $r$ $1$ . We have found the prices for both types of tickets: a regular ticket costs \$$59$, and a VIP ticket costs \$$170$.