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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineA coffee shop is having a sale on prepackaged coffee and tea. Yesterday they sold 1414 packages of coffee and 3636 packages of tea, for which customers paid a total of $386\$386. The day before, 2020 packages of coffee and 3636 packages of tea was sold, which brought in a total of $428\$428. How much does each package cost?\newlinePer package, coffee costs $\$_____ and tea costs $\$_____.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineA coffee shop is having a sale on prepackaged coffee and tea. Yesterday they sold 1414 packages of coffee and 3636 packages of tea, for which customers paid a total of $386\$386. The day before, 2020 packages of coffee and 3636 packages of tea was sold, which brought in a total of $428\$428. How much does each package cost?\newlinePer package, coffee costs $\$_____ and tea costs $\$_____.
  1. Define Variables: Let's denote the cost of one package of coffee as xx dollars and the cost of one package of tea as yy dollars. We can write two equations based on the information given for the two days.\newlineYesterday's sale: 1414 packages of coffee and 3636 packages of tea for $386\$386.\newlineThe day before's sale: 2020 packages of coffee and 3636 packages of tea for $428\$428.
  2. Write Equations: Write the equations based on the information given.\newlineFor yesterday's sale: 14x+36y=38614x + 36y = 386\newlineFor the day before's sale: 20x+36y=42820x + 36y = 428
  3. Eliminate Variable: We have the system of equations:\newline14x+36y=38614x + 36y = 386\newline20x+36y=42820x + 36y = 428\newlineWe need to eliminate one of the variables. Since the coefficients of yy are the same in both equations, we can eliminate yy by subtracting the first equation from the second.
  4. Subtract Equations: Subtract the first equation from the second to eliminate yy.\newline(20x+36y)−(14x+36y)=428−386(20x + 36y) - (14x + 36y) = 428 - 386\newline20x−14x+36y−36y=428−38620x - 14x + 36y - 36y = 428 - 386\newline6x=426x = 42
  5. Solve for x: Solve for x.\newline6x=426x = 42\newlinex=426x = \frac{42}{6}\newlinex=7x = 7\newlineSo, each package of coffee costs $7\$7.
  6. Substitute xx: Now that we have the value of xx, we can substitute it back into one of the original equations to find the value of yy. Let's use the first equation: 14x+36y=38614x + 36y = 386 Substitute x=7x = 7 into the equation: 14(7)+36y=38614(7) + 36y = 386 98+36y=38698 + 36y = 386
  7. Solve for y: Solve for y.\newline36y=386−9836y = 386 - 98\newline36y=28836y = 288\newliney=28836y = \frac{288}{36}\newliney=8y = 8\newlineSo, each package of tea costs $8\$8.

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