Question

initial discussion prompt
read the scenario below and answer any questions.

1. you are planning a party on a limited budget. in order to do so you will need to determine how much it will cost (total) to rent a space and pay per person for meals. if the cost of the rental space is $780 and the price per person for food is $9.75, how much would it cost to host 50 people for the party? in your post, please write a linear equation in the form of y = mx + b to model the scenario. give a description of what each letter stands for that makes sense for this situation. then, substitute the numbers in and solve to show the total cost for 50 people.
2. if your budget was $1500 total, how many people can you invite? write a similar equation and solve to find out how many guests you could invite with a total budget of $1500.

Explanation:

Step1: Define variables for linear equation

Let \(y\) = total cost, \(x\) = number of people. \(m = 9.75\) (cost per person), \(b=780\) (rental cost). The linear equation is \(y = 9.75x+780\).

Step2: Calculate cost for 50 people

Substitute \(x = 50\) into \(y=9.75x + 780\).
\(y=9.75\times50+780\)
\(y = 487.5+780\)
\(y=1267.5\)

Step3: Solve for number of people with budget \(y = 1500\)

Substitute \(y = 1500\) into \(y=9.75x + 780\).
\(1500=9.75x+780\)
Subtract 780 from both sides: \(1500 - 780=9.75x\)
\(720 = 9.75x\)
Divide both sides by 9.75: \(x=\frac{720}{9.75}\approx73.85\)
Since we can't invite a fraction of a person, \(x = 73\)

Answer:

1. The linear equation is \(y = 9.75x+780\) (where \(y\) is total cost, \(x\) is number of people). The cost for 50 people is \(\$1267.5\).
2. With a budget of \(\$1500\), you can invite \(73\) people.