Help with linear algebra question?

Lawns fertilizers are rated according to their “N/P/K” content. N for the percent of nitrogen, P for phosphorus, and K for potassium. So, an “8/10/5″ fertilizer has 8% nitrogen, 10 %phosphorus, and 5% potassium.

Suppose you have three types of lawn fertilizer in your garden supply store. Superlawntastic is 5/10/15, Turf-O-Matic is 10/4/2, and DynoGreenUp is 10/5/5. A groundskeeper for a golf course calls your store and says he needs 1000 kg of fertilizer rated 9.5/5.0/4.5. None of your fertilizers provide exactly what he is looking for, but you figure you can make a blend of the three that will meet his needs.

a) Set up and use a system of linear equations to determine the amount of each type of fertilizer you should use to create 1000 kg of 9.5/5.0/4.5.
b) Suppose instead the groundskeeper had ordered 1000 kg of 8/5/5. Could you do a different blend to provide that? If so, determine what blend to use; if not, explain why.

Help with setup/understand?

S = (5 10 15)
T = (10 4 2)
D = (10 5 5)

You want x1, x2 and x3 such that

x1S + x2T + x3D = (9.5 5.0 4.5)

This problem can be solved in matrix form as follows

| 5 10 10| |x1| . |9.5|
| 10 4 5 | .|x2| =|5.0|
| 15 2 5 | .|x3| . |4.5|

So what you need is the inverse of the 3X3 matrix on the LHS

The inverse is

|-0.10 0.30 -0.10|
|-0.25 1.25 -0.75|
| 0.40 -1.40 0.80|

Premultiplying by the inverse gives you

|x1| . |-0.10 0.30 -0.10| |9.5|
|x2| =|-0.25 1.25 -0.75| |5.0|
|x3| . | 0.40 -1.40 0.80| |4.5|

Now you just multiply the matrices and get

|x1| . |-0.10 0.30 -0.10| |9.5|
|x2| =|-0.25 1.25 -0.75| |5.0|
|x3| . | 0.40 -1.40 0.80| |4.5|

|x1| . |0.1|
|x2| =|0.5|
|x3| . |0.4|

So 10% of S (100 kg) 50% of T (500 kg) and 40% D (400 kg)

Kierland Golf Club at Westin Kierland Resort in Scottsdale

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