Question

How do I set this up?

Pastaroni is a local Italian restaurant that serves a specialized pasta salad. Included in this salad is macaroni, linguine and bow-tie pasta. The salad contains six times as much linguine as it does macaroni. Both macaroni and linguine sell for $4; bow-tie pasta sells for $8. If the restaurant makes 140 pounds of pasta every day, how many pounds of each type of pasta are needed if the salad sells for $6?

Answer 1

Express it as a system of linear equations: one is the sum of the weights of the pasta (linguine,macaroni, and bow-tie pasta), one is expressing the fact that one macaroni is equal to 6 linguine,and the other is an equation equating the sum of the prices of the salad components with the price of the salad itself.

let x be the weight of macaroni, y be the weight of the linguine, and z be the weight of the bow-tie pasta
\[x+y+z=140\]\[x - 6y\ \ \ \ \ = 0\]\[4x + 4y + 8z = 140*6\]

Answer 2

Thank you. If I have any further questions, can I ask you?

Answer 3

simply use gaussian elimination http://en.wikipedia.org/wiki/Gaussian_elimination to solve the system, and you will get x = 60, y = 10, and z = 70

you can ask anybody here :-D

Answer 4

Thank You so much again.

Answer 5

Which order do I solve it in?

Answer 6

They only gave me the x+y=# not the three equations thats why i am lost

Answer 7

something wrong in my answer: I mixed up the macaroni and the linguine, the equation up there that says x - 6y = 0 should by 6x - y = 0 to correctly show the fact that there is 6 times more linguine in the salad than macaroni

the final answer should be x = 10, y = 60, and z = 70

Answer 8

Lets say i set it up like this

\[x+y+z=140\]
\[6x-y=0\]
\[4x+4y+8z=140(6)\]

do i solve the first and second and depending on my answer plug it in to the third or what?

Answer 9

you don't necessarily have to solve for each individual equation, if you are using elimination. If you are using substitution then you have to pick something to solve ( the second equation is very easy to solve for y, for example, so you solve it for y so that you can plug 6x instead of y in the other equations)

Answer 10

@Mrbonez321:You can solve them as any order you want, it doesn't matter as long you are not making any mistake.

Answer 11

Thank you both!

Answer 12

http://www.wolframalpha.com/input/?i=rref%28%28%281%2C1%2C1%2C140%29%2C%286%2C-1%2C0%2C0%29%2C%284%2C4%2C8%2C140*6%29%29%29

Answer 13

That last link lost me. Sorry

Answer 14

Thank you agdgdgdgwngo I just solved it by hand using your help. Thank you for spending this much time helping me. I really appreciate it!