Question

Will mark brainlest
Which System of Inequalities could be used to solve the problem below?

A local theater has a maximum capacity of 1,000 people. For their latest production, adult tickets cost $25 and youth tickets cost $12.50. The theater must bring in more than $15,000 for the show to go on. How many of each tickets must be sold to save the show?

Answer 1

Answer:

200 adult tickets and 800 youth tickets must be sold to save the show

Step-by-step explanation:

Systems of Linear Equations

If 2 or more variables are related in 2 or more equations, we have a system which could provide a solution by a number of methods.

The audience of the local theater could have x adults and y youth people. It has a maximum capacity of 1,000 people. It can be expressed as

$\displaystyle x+y\leq 1000$

Since adult tickets cost $25 and youth tickets cost $12.50, the total income for selling the tickets is 25x+12.5y and it should be more than $15,000, so

$\displaystyle 25x+12.50y\geq15,000$

Multiplying by 2

$\displaystyle 50x+25y\geq30,000$

Dividing by 25

$\displaystyle 2x+y\geq1200$

If the theater is at maximum capacity and they want to save the show, the following system of equations must be solved  

$\displaystyle \left\{\begin{matrix}x+y=1000\\ 2x+y=1200\end{matrix}\right.$

Multiplying the first equation by 2

$\displaystyle \left\{\begin{matrix}-2x-2y=-2000\\ 2x+y=1200\end{matrix}\right.$

Adding both equations and solving for x

$\displaystyle -y=-800$

$\displaystyle y=800$

We know that

$\displaystyle x+y=1000$

Solving for x

$\displaystyle x=1000-800$

$\displaystyle x=200$

200 adult tickets and 800 youth tickets must be sold to save the show