Question

The local orchestra has been invited to play at a festival. There are 111 members of the orchestra and 6 are licensed to drive large multi-passenger vehicles. Busses hold 25 people but are much more expensive to rent. Passenger vans hold 12 people. make a system of equations to find the smallest number of busses the orchestra can rent. Let x= the number of busses and y= the number of vans make an equation representing the number of vehicles needed. __________ make an equation representing the total number of seats in vehicles for the orchestra members. __________ Solve the system of equations. How many busses does the orchestra need to rent? ____________ How many 12-passenger vans does the orchestra need to rent? ____________

Answer 1

- Make an equation representing the number of vehicles needed.

We have six drivers so

x + y ≤ 6

That's not really an equation; it's an inequality.  We want to use all our drivers so we can use the small vans, so

x + y = 6

- Make an equation representing the total number of seats in vehicles for the orchestra members.

s = 25x + 12y

That's how many seats total; it has to be at least 111 so again an inequality,

25x + 12y ≥ 111

We solve it like a system of equations.  

x + y = 6

y = 6 - x

111 = 25x + 12y = 25x + 12(6-x)

111 = 25x + 72 - 12x

111 - 72 = 13 x

39 = 13 x

x = 3

Look at that,  it worked out exactly.  It didn't have to.

y = 6 - x = 3

Answer: 3 buses, 3 vans

Answer 2

Using a System of equations, the number of buses and vans required are 3 respectively.

Using the system of equations :

Total number of buses :

• b + v ≤ 6 - - - - (1)

Total number of passengers :

• 25b + 12v ≥ 111 - - - - (2)

From (1)

• b = 6 - v - - - - - (3)

Substitute (3) into (2)

25(6-v) + 12v = 111

150 - 25v + 12v = 111

-13v = 111 - 150

-13v = - 39

v = 39/13

v = 3

From (3)

b = 6 - 3

b = 3

Hence, the number of vans and buses required are 3 and 3 respectively.

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