Question

A florist is creating 10 centerpieces. Roses cost $2.50 each, lilies cost $4 each, and irises cost $2 each. A customer plans on spending $300 on 10 centerpieces with each centerpiece containing 12 flowers, with twice as many roses as the number of irises and lilies combined.

a) Write a system of linear equations that represents the situation. Then write a matrix equation that corresponds to your system.

b) Find the number of flowers of each type that the florist can use to create the 10 centerpieces.

Answer 1

Answer:

8 Roses; 1 lilie ; 3 irises

Step-by-step explanation:

Let the number of roses be 'a' , number lilies be 'b' and number of irises be 'c'

$300 for 10 centerpieces means $30 per centerpiece. therefore,

$2.5a+4b+2c = 30$ .......................(1)

also each centerpiece must contain 12 flowers, therefore,

$a+b+c=12$ .................................(2)

also twice as much roses as the number of irises and lilies combined

$a = 2(b+c)$

$a-2b-2c=0$ ............................... (3)

a) System of equation are equations (1) , (2) & (3).

Writing a matrix equation

$\left[\begin{array}{ccc}2.5a&4b&2c\\a&b&c\\a&-2b&-2c\end{array}\right] = \left[\begin{array}{ccc}30\\12\\0\end{array}\right]$

b) Solving the simultaneous equation,

from (3), $a=2b+2c$

substituting for 'a' in equation (1) and (2),

⇒ 2.5(2b+2c)+4b+2c=30

⇒ 5b + 5c + 4b + 2c = 30

⇒ 9b + 7c = 30 ..................... (4)

⇒ 2b + 2c + b + c = 12

⇒ 3b + 3c = 12

⇒ b = 4 - c ..............................(5)

Substituting (5) in (4)

9 (4 - c) + 7c = 30

36 - 9c+7c =30

36-30 = 2c

c = 3

b = 4-3 = 1

a = 2(3+1) = 8