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At a concession​ stand, fivefive hot dog left parenthesis s right parenthesishot dog(s) and fourfour hamburger left parenthesis s right parenthesishamburger(s) cost ​$16.5016.50​; fourfour hot dog left parenthesis s right parenthesishot dog(s) and fivefive hamburger left parenthesis s right parenthesishamburger(s) cost ​$17.2517.25. find the cost of one hot doghot dog and the cost of one hamburgerhamburger.

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nobillionaireNobley
Let the cost of one hot dog be x and the cost of one hamburger be y, then given that five hot dog(s) and four hamburger(s) cost ​$16.50, we have

5x + 4y = 16.50 . . . (1)

Given​ that four hot dog(s) and fivehamburger(s) cost ​$17.25, we have

4x + 5y = 17.25 . . . (2)

(1) x 4 ⇒ 20x + 16y = 66 . . . (3)
(2) x 5 ⇒ 20x + 25y = 86.25 . . . (4)

(3) - (4) ⇒ -9y = -20.25
⇒ y = -20.25 / -9 = 2.25

Substituting for y into (1) gives:

5x + 4(2.25) = 16.50
⇒ 5x + 9 = 16.50
⇒ 5x = 16.50 - 9 = 7.5
⇒ x = 7.5 / 5 = 1.5

Therefore, the cost of one hot dog is $1.50 and the cost of one hamburger is $2.25.