Question

A deli sells three sizes of turkey sandwiches: the small turkey sandwich contains 4 ounces of meat and sells for $3.00; the regular turkey sandwich contains 9 ounces of meat and sells for $3.50; and the large turkey sandwich contains 12 ounces of meat and sells for $4.00. A customer request a selection of each size for a reception. She and the manager agree on a combination of 40 sandwiches made from 19 pounds 6 ounces of turkey for a total cost of $137. How many of each size sandwich will be in this combination?

Answer 1

Answer choices:

20 small sandwiches, 16 medium sandwiches, 14 large sandwiches.

18 small sandwiches, 22 medium sandwiches, 10 large sandwiches.

16 small sandwiches, 28 medium sandwiches, 6 large sandwiches.

14 small sandwiches, 18 medium sandwiches, 18 large sandwiches.

Answer 2

three unknowns, so three equations will be needed

let's let S = # of small sandwiches, M = # of mediums, L = # of larges

since she wants 40 sandwiches total, S + M + L = 40

Answer 3

now, we can write another equation for the total amount of meat. 1 lb = 16 oz, so:

19 lb 6 oz = 19*16 + 6 = 310oz of meat

each small is 4 oz of meat, so the amount of meat from all the small sandwiches (S) = 4S
same logic with the mediums (9 oz) and larges (12 oz): 9M and 12L respectively

therefore 4S + 9M + 12L = 310

Answer 4

last equation: the price

since each small is $3.00, the total cost of (S) small sandwiches is 3S
same logic with the mediums and larges: 3.5M + 4L

so 3S + 3.5M + 4L = 137

Answer 5

putting everything together:
S + M + L = 40
4S + 9M + 12L = 310
3S + 3.5M + 4L = 137

best way is to try and reduce this system of 3 to 2 equations. you can multiply the first equation by 4, for example, to get 4S + 4M + 4L = 160. this along with 4S + 9M + 12L = 310
---> you can subtract these two equations to eliminate S, so you're left with an equation with M and L only

you can repeat this logic with the first and third equations, multiplying the first eqn by 3 to eliminate S again, and produce another equation in terms of M and L only.

from there, you can eliminate either M or L using the same method from before, reducing the equation to one variable. once you have one variable, you can plug it into the original system, reduce the system to two variables again, and solve via elimination again. keep going until you have all variables.

when you're done, it would be best to substitute your S M and L values back into the three equations to make sure everything is consistent