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A minor league baseball park has 7000 seats. Box seats cost $6, grandstand seats cost $4, and bleacher seats
cost $2. When all seats are sold, the revenue is $26,400. If the number of box seats is one-third the number of
bleacher seats, how many seats of each type are there? Give a complete sentence explaining what your answers
mean.



Answer :

CintiaSalazar

Taking into account the definition of a system of linear equations, the number of box seats is 400, the number of bleacher seats is 1200 and the number of grandstand is 5400.

System of linear equations

A system of linear equations is a set of two or more equations of the first degree, in which two or more unknowns are related.

Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are satisfied. That is, the values ​​of the unknowns must give the solution proposed in both equations.

Amount of seats of each type

In this case, a system of linear equations must be proposed taking into account that:

  • "B" is the number of box seats.
  • "G" is the numer of grandstand seats.
  • "BL" is the number of bleacher seats.

You know that:

  • A minor league baseball park has 7000 seats.
  • Box seats cost $6, grandstand seats cost $4, and bleacher seats cost $2. When all seats are sold, the revenue is $26,400.
  • The number of box seats is one-third the number of bleacher seats,

So, the system of equations to be solved is

Equation 1: B + G + BL= 7000

Equation 2: 6B + 4G + 2BL= 26400

Equation 3: B= 1/3BL

There are several methods to solve a system of equations, it is decided to solve it using the substitution method, which consists of clearing one of the two variables in one of the equations of the system and substituting its value in the other equation.

In this case, replacing equation 3 in equation 1 and isolating the variable G, you get:

1/3BL + G + BL= 7000

4/3BL + G= 7000

G= 7000 - 4/3BL

Substituting this expression and equation 3 into equation 2 you get

6×1/3BL + 4×(7000 - 4/3BL) + 2BL= 26400

Solving:

2BL + 4×7000 - 4×4/3BL + 2 BL= 26400

2BL + 28000 - 16/3BL + 2 BL= 26400

2BL - 16/3BL + 2 BL= 26400 - 28000

-4/3BL= -1600

BL= (-1600)÷ (-4/3)

BL= 1200

Remembering that B= 1/3BL and G= 7000 - 4/3BL and replacing the value of BL you get:

B= 1/3BL= 1/3×1200 → B= 400

G= 7000 - 4/3BL= 7000 - 4/3×1200 → G= 5400

In summary, the number of box seats is 400, the number of bleacher seats is 1200 and the number of grandstand is 5400.

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