Question

1. There were 36,000 people at a horse race in Lexington, Kentucky. The day's

receipts were $250,000. The only two types of seats available were clubhouse
or grandstand seats. How many people paid $12.00 for clubhouse seats and
how many people paid $5.00 for grandstand seats? Only an algebraic solution
will earn credit. State what any variables represent by writing a "let statement”.​

Answer 1

• 10000 people paid $12.00 each for clubhouse seats and
• 26000 people paid $5.00 each for grandstand seats.

Explanation:

The question is asking for a system of equations, which make explanations easy. :)

Define the variables. Setting
`x` to the number of clubhouse seats sold and
`y` to the number of grandstand seats sold will be sufficient. The "let statement[s]" will be:

• Let
  `x` be the number of clubhouse seats sold.
• Let
  `y` be the number of grandstand seats sold.

The number of equations shall be no less than the number of variables for the solution to be unique. There are two variables. It will take at least two equations to find a unique solution.

Everyone at the race need a seat. The number clubhouse seats plus the number of grandstand seats shall be the same as the number people at the race. There were 36,000 people. Therefore the first equation shall be:

`x + y = 36000`.

Every clubhouse seat will add $12.00 to the receipt.
`x` clubhouse seats will add $
`12\;x` to the receipt. Similarly,
`y` grandstand seats will add $
`5\;y` to the receipt. The two values shall add up to $250,000.

Drop the dollar sign to get the second equation:

`12\;x +5\;y =250000`.

Hence the system:

`\displaystyle \left\{\begin{aligned}& x + y = 36000 && \textcircled{\raisebox{-0.9pt}1}\\ & 12\;x + 5\;y = 250000 && \textcircled{\raisebox{-0.9pt}2}\end{aligned} \phantom{\small credit for the raisebox hack: tex[dot]stackexchange[dot]com/questions/7032/good-way-to-make-textcircled-numbers}`.

Solve this system.

The first non-zero coefficient in equation
`\textcircled{\raisebox{-0.9pt}1}` is already one. That's the coefficient for
`x`. Use multiples of equation
`\textcircled{\raisebox{-0.9pt}1}` to get rid of
`x` in other equations (equation
`\textcircled{\raisebox{-0.9pt}2}` in this case.)

`-12` times equation
`\textcircled{\raisebox{-0.9pt}1}` is

`-12 \;x - 12\;y = -432000`.

Add
`-12* \textcircled{\raisebox{-0.9pt}1}` to
`\textcircled{\raisebox{-0.9pt}2}` to get:

`0\;x + -7\;y = -182000`.

Divide both sides by -7 to get:

`y = 26000`.

Add -1 times this equation to equation
`\textcircled{\raisebox{-0.9pt}1}` to get:

`x = 10000`.

That is:

`\displaystyle \left\{\begin{aligned}&x = 10000\\&y = 26000\end{aligned}`.

In other words,

• 10000 clubhouse seats were sold, and
• 26000 grandstand seats were sold.