Question

a catering company is setting up for a high school graduation party. they expect 150 people to attend. they can provide small tables that seat 6 people and large tables that seat 10 people.
a. find a combination of small and large tables that seats exactly 150 people.
b. let x represent the number of small tables and y represent the number of large tables. write an equation to represent the relationship between x and y.
c. explain what the point (20,5) means in this situation.
d. is the point (20,5) a solution to the equation you wrote? explain your reasoning.

Explanation:

Step1: Find combination for part a

Let's assume the number of small tables \(x\) and large tables \(y\). We know \(6x + 10y=150\).
If \(x = 0\), then \(10y=150\), \(y = 15\).
If \(y=0\), then \(6x = 150\), \(x = 25\).
Another combination: Let \(x = 10\), then \(6\times10+10y=150\), \(60 + 10y=150\), \(10y=90\), \(y = 9\).

Step2: Write equation for part b

Since each small table seats \(6\) people and each large table seats \(10\) people, and total number of people is \(150\). The equation is \(6x+10y = 150\).

Step3: Explain the point for part c

In the context of the problem, \(x\) represents the number of small tables and \(y\) represents the number of large tables. So the point \((20,5)\) means there are \(20\) small tables and \(5\) large tables.

Step4: Check if it's a solution for part d

Substitute \(x = 20\) and \(y = 5\) into the equation \(6x+10y\).
\(6\times20+10\times5=120 + 50=170
eq150\)

Answer:

a. One combination is \(10\) small tables and \(9\) large tables (or other valid combinations like \(0\) small and \(15\) large, \(25\) small and \(0\) large).
b. \(6x + 10y=150\)
c. It means \(20\) small tables and \(5\) large tables.
d. No, because when \(x = 20\) and \(y = 5\), \(6x+10y=170
eq150\)