A Broadway theater has 500 seats, divided into orchestra,...

A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $$\$ 50,$$ main seats for $$\$ 35,$$ and balcony seats for $$\ $25$$. If all the seats are sold, the gross revenue to the theater is $$\$ 17,100 .$$ If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $$\$ 14,600$$. How many of each kind of seat are there?

Key Concepts

System of Linear Equations

This concept involves setting up multiple equations based on the relationships described in a problem, where each equation represents a constraint or condition involving unknown variables. In overall algebra and problem solving, such systems allow one to solve for unknowns by using methods like substitution or elimination when several linear relations must be satisfied simultaneously.

Constraint Equations

Constraint equations are the mathematical expressions that represent limitations or fixed totals in a problem. In many contexts, including allocation or budgeting problems, the sum of variables must meet given totals, ensuring that the solution adheres to the physical or logical limits described in the scenario.

Weighted Sum Calculations

Weighted sum calculations occur when different components contribute unequally to an overall total. This concept is particularly important in revenue or cost calculations where each item has an associated weight or price, and the overall total is computed by summing the products of these weights and the corresponding quantities.

Handling Partial or Fractional Values in Equations

Many real-world problems involve scenarios where only a fraction of a quantity contributes to an outcome. This concept deals with correctly incorporating fractional participation (such as selling only a part of the available items) into the equations, thereby modifying the original constraints and requiring careful adjustment of the calculations.

Transcript

00:01 The topic of this question is systems of linear equations.

00:06 This question is about a broadway theater with 500 seats that has three types of seats.

00:14 Its orchestra seats sell for $50 each.

00:19 The main seats are $35 each, and the balcony seats are $25 each.

00:30 Given that when all the seats are sold out, their revenue is $17.

00:37 $11 ,100 and that when all the seats except for half of the orchestra seats are sold out, the revenue is $14 ,600, we want to find the number of orchestra, main, and balcony seats in this theater.

00:59 Okay, so basically to start answering this question, we know we need to find the values of three numbers, that is the number of orchestra seats.

01:09 The number of main seats, and the number of balcony seats.

01:16 I've named those based on the names of the types of seats.

01:22 And so we can call these three numbers r, m, and b.

01:27 Those are the variable names.

01:32 And to find their numerical values, we need three equations involving those variables to solve for.

01:41 So first of all, one simple equation that we can get from the information is, from the total number of seats.

01:49 So the number of orchestra seats plus the number of main and balcony seats is 500.

01:55 And so simply put, the sum of our three values is 500.

02:07 Now the sum of the sales from all these 500 seats is this number here.

02:15 Now, how do we express that in terms of r, b, and m? well, we know that this is the number, this number is the total of sales from r seats, from m seats, and from b seats.

02:31 So the total sales from r seats, that would be $50 per seat times r seats.

02:40 That's $50 r dollars altogether.

02:43 Similarly, the sales from the main seats is 35 times m, and the total sales from balcony seats is 25 times b.

02:58 Now, if all the main and balcony seats are sold and half of the orchestra seats, we have this much.

03:08 And so we still have the same amount of revenue from main and balcony seats, but this time we're only adding in half of the orchestra seats...

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