I am having trouble setting this one up

Question

A florist is creating 10 centerpieces for the tables at a wedding reception. Roses cost $2.50 each, lilies cost $4 each, and irises cost $2 each. The customer has a budget of $370 allocated for the centerpieces and wants each centerpiece to contain 15 flowers, with twice as many roses as the number of irises and lilies combined. (Let r represent the number of roses, l represent the number of lilies, and i represent the number irises. )

 

(a) Write a system of linear equations that represents the situation.



_____________   = 150

_____________   = 370

_____________    = 0


(b) Write a matrix equation that corresponds to your system.

(c) Solve your system of linear equations using an inverse matrix. Find the number of flowers of each type that the florist can use to create the 10 centerpieces.

roses

lilies

irises

Answer 1

Blake,

I hope this is time to do you some good.  Two of the equations are straight

forward from the info provided. The third is only found by manipulating some

of the info.

 

r + l + i = 150         the number of each type of flower added together must equal the total number of flowers.

 

2.5r + 4l + 2i = 370     the cost of each type of flower must add up to the total budget

 

The third equation as given in the problem would be r = 2(l + i). But to line things up

for the matrix, we manipulate it to be:

 

r - 2l - 2i = 0

 

So our matrix equation becomes:

 

1     1     1          r          150

2.5   4    2     *    l    =    370

1    -2    -2          i             0

 

                     or

 

        A          *    X    =     B

 

AX = B can of course be manipulated to X = A^-1B, where X is our matrix of values

for each variable.  Now we use an inverse matrix calculator to find:

 

         A^-1         *       B          =       X

 

2/3     0      1/3         150             100 roses

-7/6  1/2   -1/12        370               10 lilies

3/2  -1/2   -1/4            0                40 irises

 

 

I'm sorry there is no good way to show a matrix in this editor. I hope it comes out legible.

 

 

Answer 2

a)

Let's first start with how many flowers we'll be dealing with. A florist wants to create 10 centerpieces, each with 15 flowers. So the grand total of flowers the florist will be handling is

 

10 centerpieces * 15 flowers/ centerpiece = 150 flowers

 

If

 

R = # roses

L = # lilies

I = # irises

 

then we know that R + L + I = 150.

 

Now that we have an equation for the amount of flowers, let's set up an equation for the cost.

 

We know the cost of a rose is $2.50.

 

C_R = $2.50

 

The cost of a lily is $4.

 

C_L = $4

 

The cost of an iris is $2.

 

C_I = $2.

 

The total budget the florist has to work with is $370. This will account for the cost of each flower times the number of each flower bought.

 

C_R*R + C_L*L + C_I*I = $370

 

Substitute your values:

 

2.50*R + 4.00*L + 2.00*I = $370

 

The equation above is what your second equation will state.

 

The third equation will be pulled from the ratio of flowers needed for each centerpiece. The florist wants twice as many roses as there are lilies and irises combined. Another way to put that is, "The number of roses will equal 2 times the quantity of lilies and irises."

 

In math that translates to:

 

R = 2*(L + I)

 

R = 2*L + 2*I

 

So

 

R - 2*L - 2*I = 0

 

This is your third equation.

 

So your system is:

 

R + L + I = 150

2.50*R + 4.00*L + 2.00*I = $370

R - 2*L - 2*I = 0

 

Three equations, three unknowns.

 

b)

In matrix form it'll be:

 

1        1       1       150

2.50   4.00   2.00   370

1       -2      -2       0

 

c)

The inverse form is tough to type out over this medium. It's all set up for you to execute though.

 

You should have 100 roses, 10 lilies, and 40 irises.

 

With systems, always check the results:

 

2.50*100 + 4.00*10 + 2.00*40 = 250 + 40 + 80 = 370. Check.

 

100 = 2*(10+40). Check.

 

100+40+10 = 150. Checkmate.

Answer 3

(a)

The total amount of money spent on each flower is the cost per flower multiplied by the number of flowers: 2.5R, 4L, 2I. The budget is $370, so the sum of the three individual flower costs must add up to this:

 

2.5R+4L+2I=370.

 

If there are 10 centerpieces, each with 15 flowers, then there is a total of 150 flowers. The number of roses plus the number of lilies plus the number of irises must sum to 150:

 

R+L+I=150.

 

Finally, with twice as many roses as irises and lilies combined, we get an equation that looks like:

 

R=2(L+I)

or

R-2L-2I=0.

================================================

(b)

Putting this into a matrix equation:

 

[1      1     1]  [R]       [150]

[2.5   4     2]  [L]   =  [370 ]

[1     -2   -2]  [I]        [  0  ]

=================================================

(c)

The matrix can be inverted in a calculator or using any kind of computer software (for example http://www.wolframalpha.com/widgets/view.jsp?id=35f68681262e42ea89b0834caa51635b)

 

Multiplying the inverse matrix by the column vector on the right hand side of the matrix equation will give you a 3 by 1 column vector for [R, L, I]