Question

Three bouquets of flowers are ordered at a florist. 3 roses, 2 carnations, and 1 tulip cost $14. 6 roses, 2 carnations, and 6 tulips cost $38. 1 rose, 12 carnations, and 1 tulip cost $18. How much does each item cost?

Answer 1

cost of 1 rose = $ 3

cost of 1 carnation = $ 1

cost of 1 tulip = $ 3

Solution:

Let "r" be the cost of 1 rose

Let "c" be the cost of 1 carnation

Let "t" be the cost of 1 tulip

3 roses, 2 carnations, and 1 tulip cost $14

So we can frame a equation as:

3 roses x cost of 1 rose + 2 carnations x cost of 1 carnation + 1 tulip x cost of 1 tulip = $ 14

`3 * r + 2 * c + 1 * t = 14`

3r + 2c + 1t = 14 ----- eqn 1

6 roses, 2 carnations, and 6 tulips cost $38

So we can frame a equation as:

6 roses x cost of 1 rose + 2 carnations x cost of 1 carnation + 6 tulip x cost of 1 tulip = $ 38

`6 * r + 2 * c + 6 * t = 38`

6r + 2c + 6t = 38 ------ eqn 2

1 rose, 12 carnations, and 1 tulip cost $18

So we can frame a equation as:

1 rose x cost of 1 rose + 12 carnations x cost of 1 carnation + 1 tulip x cost of 1 tulip = $ 18

`1 * r + 12 * c + 1 * t = 18`

r + 12c + t = 18 ----- eqn 3

Let us solve eqn 1 and eqn 2 and eqn 3 to find values of "r" "c" "t"

3r + 2c + 1t = 14 ----- eqn 1

6r + 2c + 6t = 38 ------ eqn 2

r + 12c + t = 18 ----- eqn 3

From eqn 1,

3r = 14 - 2c - t

`r = (14 - 2c - t)/(3)`

Substitute the above value of r in eqn 2

`6((14 - 2c - t)/(3))+ 2c + 6t = 38\\\\28 - 4c - 2t + 2c + 6t = 38\\\\-2c +4t = 10\\\\-2c = 10 - 4t\\\\2c = 4t - 10\\\\c = 2t - 5`

Substitute c = 2t - 5 and
`r = (14 - 2c - t)/(3)` in eqn 3

`12(2t - 5) + (14 - 2c - t)/(3) + t = 18\\\\24t - 60 + (14-2(2t - 5) - t)/(3) + t = 18\\\\72t - 180 + 14 - 4t +10 - t + 3t = 54\\\\70t = 54 + 180 - 14 -10\\\\70t = 210\\\\t = 3`

t = 3

Substitute t = 3 in c = 2t - 5

c = 2(3) - 5

c = 1

Substitute t = 3 and c = 1 in
`r = (14 - 2c - t)/(3)`

`r = (14 - 2(1) - 3)/(3)\\\\r = (14 - 2 - 3)/(3)\\\\r = (9)/(3) = 3`

r = 3

Summarizing the results:

cost of 1 rose = $ 3

cost of 1 carnation = $ 1

cost of 1 tulip = $ 3