Question

7
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 that small tables seat \(6\) people and large tables seat \(10\) people. We want \(6x + 10y=150\). Let's try \(x = 10\) and \(y = 9\). Then \(6\times10+10\times9=60 + 90=150\)

Step2: Write equation for part b

Since small tables seat \(6\) people (\(x\) is the number of small tables) and large tables seat \(10\) people (\(y\) is the number of large tables) and the total number of people is \(150\), the equation is \(6x + 10y=150\)

Step3: Explain point \((20,5)\) for part c

The \(x -\)value is the number of small tables and the \(y -\)value is the number of large tables. So the point \((20,5)\) means \(20\) small tables and \(5\) large tables

Step4: Check if \((20,5)\) is a solution for part d

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

Answer:

a. \(10\) small tables and \(9\) large tables (other combinations are possible)
b. \(6x + 10y=150\)
c. \(20\) small tables and \(5\) large tables
d. No, because when \(x = 20\) and \(y = 5\), \(6x+10y=170
eq150\)