One day, a vendor sold 58 hot dogs and 84 bratwursts for a total of $268.30. The next day, the vendor sold 44 hot dogs and 92 bratwursts for a total of $271.40. What is the cost of one hot dog and the cost of one bratwurst? If the Winchester family buys 3 hot dogs and 2 bratwursts, how much will they pay?

Question

One day, a vendor sold 58 hot dogs and 84 bratwursts for a total of  $268.30. The next day, the vendor sold 44 hot dogs and 92 bratwursts for a total of  $271.40. What is the cost of one hot dog and the cost of one bratwurst? If the Winchester family buys 3 hot dogs and 2 bratwursts, how much will they pay?

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Equations setup: Let's denote the cost of one hot dog as H dollars and the cost of one bratwurst as B dollars. On the first day, the vendor sold 58 hot dogs and 84 bratwursts for a total of $268.30. This gives us the equation: 58H + 84B = 268.30Second day sales: On the second day, the vendor sold 44 hot dogs and 92 bratwursts for a total of $271.40. This gives us the second equation: 44H + 92B = 271.40Elimination method: We now have a system of two equations with two variables: 1) 58H + 84B = 268.30 2) 44H + 92B = 271.40 We can solve this system using the method of substitution or elimination. Let's use the elimination method to find the values of H and B.Multiplication for elimination: First, we'll multiply the first equation by 44 and the second equation by 58 to align the coefficients of H for elimination: (58H + 84B) * 44 = 268.30 * 44 (44H + 92B) * 58 = 271.40 * 58Subtracting equations: After performing the multiplication, we get: 2552H + 3696B = 11805.20 2552H + 5336B = 15741.20Solving for B: Now, we'll subtract the second equation from the first to eliminate H: (2552H + 3696B) - (2552H + 5336B) = 11805.20 - 15741.20Substitute back for H: This simplifies to: -1640B = -3936Calculating total cost: Dividing both sides by -1640 to solve for B, we get: B = -3936 / -1640 B = 2.40 So, the cost of one bratwurst is $2.40.Now that we have the value of B, we can substitute it back into one of the original equations to find H. Let's use the first equation: 58H + 84(2.40) = 268.30Calculating the value of 84 times 2.40: 58H + 201.60 = 268.30Subtracting 201.60 from both sides to solve for H: 58H = 268.30 - 201.60 58H = 66.70Dividing both sides by 58 to find the value of H: H = 66.70 / 58 H = 1.15 So, the cost of one hot dog is $1.15.Now that we have the cost of one hot dog (H = $1.15) and one bratwurst (B = $2.40), we can calculate the total cost for the Winchester family, who buys 3 hot dogs and 2 bratwursts: Total cost = 3H + 2B Total cost = 3(1.15) + 2(2.40)Calculating the total cost: Total cost = 3.45 + 4.80 Total cost = 8.25 The Winchester family will pay $8.25 for 3 hot dogs and 2 bratwursts.

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