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Expert-verified Found in: Page 22 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Kyle is getting some flowers for Olivia, his Valentine. Being of a precise analytical mind, he plans to spend exactly $24 on a bunch of exactly two dozen flowers. At the flower market they have lilies ($3 each), roses ($2 each), and daisies ($0.50 each). Kyle knows that Olivia loves lilies; what is he to do?

Kyle would buy bouquet of $24$ consisting of $3$ lilies, $3$ roses and $18$ daisies.

See the step by step solution

## Step 1: Represent the data in terms of equations.

Let ‘l’ represents the number of lilies, ‘r’ represents the number of roses and ‘d’ represents the number of daisies.

The sum of number of flowers should be equal to .$24$

The equations are,

$\begin{array}{c}l+r+d=24\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}......\left(1\right)\\ 3l+2r+0.5d=24\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}......\left(2\right)\end{array}$

Perform the operation on equations, $3×\left(1\right)-\left(2\right)$

$\begin{array}{l}3l+3r+3d=72\\ 3l+2r+0.5d=24\\ -----------\\ r=48-2.5d\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}......\left(3\right)\end{array}$

Perform the operation on equations,$2×\left(1\right)-\left(2\right)$ .

$\begin{array}{l}2l+2r+2d=48\\ 3l+2r+0.5d=24\\ -----------\\ l=-24+1.5d\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}......\left(4\right)\end{array}$

## Step 2: Find the values of the variables.

Choose the value of ‘d’ such that the values of ‘l’ and ‘r’ are positive.

Using trial and error method, the value should be, $d=18$

Substitute this value in the equations (3) and (4)

$\begin{array}{c}r=48-2.5\left(18\right)\\ \therefore r=3\end{array}$

$\begin{array}{l}l=-24+1.5\left(18\right)\\ \therefore l=3\end{array}$

Verification:

$\begin{array}{c}l+r+d=24\\ ⇒3+3+18=24\\ \therefore 24=24\end{array}$

The number of lilies, roses and daisies are,$l=3,r=3,d=18$ .

Hence, Kyle would buy a bouquet of $24$ consisting of $3$lilies, $3$ roses and $18$daisies. ### Want to see more solutions like these? 