Question: Write and solve a system of equations for the following problem: A concert hall has 1,250 seats.
Tickets are sold for $55 and $70. If all seats are sold, how many seats of each type should be sold to
generate a revenue of $80,000?
Enter your solutions below. Round the nearest whole number if necessary.
There should be _____ of the $55 seats and _____ of the $70 seats.
1. **State the problem:**
We have a concert hall with a total of $1250$ seats.
Tickets are sold at two prices: $55$ and $70$.
We want to find how many tickets of each price are sold to generate a total revenue of $80000$.
2. **Define variables:**
Let $x$ be the number of $55$-dollar tickets sold.
Let $y$ be the number of $70$-dollar tickets sold.
3. **Write the system of equations:**
Since all seats are sold:
$$x + y = 1250$$
The total revenue is:
$$55x + 70y = 80000$$
4. **Solve the system:**
From the first equation:
$$y = 1250 - x$$
Substitute into the second equation:
$$55x + 70(1250 - x) = 80000$$
5. **Simplify and solve for $x$:**
$$55x + 87500 - 70x = 80000$$
$$55x - 70x = 80000 - 87500$$
$$-15x = -7500$$
Divide both sides by $-15$:
$$\cancel{-15}x / \cancel{-15} = \frac{-7500}{-15}$$
$$x = 500$$
6. **Find $y$:**
$$y = 1250 - 500 = 750$$
7. **Answer:**
There should be **500** tickets sold at $55$ and **750** tickets sold at $70$.