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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineAngie wanted to stock up on drinks for an upcoming party. First, she spent $44\$44 on 1212 cases of juice and 1010 cases of soda, which is all the store had in stock. A few days later, she returned to the store and purchased an additional 1212 cases of juice and 1313 cases of soda, spending a total of $50\$50. What is the price of each drink?\newlineThe price for a case of juice is $____\$\_\_\_\_, and the price for a case of soda is $_________\$\_\_\_\_\_\_\_\_\_.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineAngie wanted to stock up on drinks for an upcoming party. First, she spent $44\$44 on 1212 cases of juice and 1010 cases of soda, which is all the store had in stock. A few days later, she returned to the store and purchased an additional 1212 cases of juice and 1313 cases of soda, spending a total of $50\$50. What is the price of each drink?\newlineThe price for a case of juice is $____\$\_\_\_\_, and the price for a case of soda is $_________\$\_\_\_\_\_\_\_\_\_.
  1. Forming Equations: Let's denote the price of a case of juice as JJ dollars and the price of a case of soda as SS dollars. We can write two equations based on the information given:\newlineFirst purchase: 12J+10S=4412J + 10S = 44\newlineSecond purchase: 12J+13S=5012J + 13S = 50\newlineWe will use these two equations to form a system of equations that we can solve using the elimination method.
  2. Elimination Method: To use the elimination method, we want to eliminate one of the variables by subtracting one equation from the other. Since the coefficient of JJ is the same in both equations, we can subtract the first equation from the second to eliminate JJ.\newlineSubtracting the first equation from the second gives us:\newline(12J+13S)−(12J+10S)=50−44(12J + 13S) - (12J + 10S) = 50 - 44
  3. Solving for S: Performing the subtraction, we get:\newline12J+13S−12J−10S=50−4412J + 13S - 12J - 10S = 50 - 44\newlineThis simplifies to:\newline3S=63S = 6
  4. Substitute S: Now we can solve for S by dividing both sides of the equation by 33:\newline3S3=63\frac{3S}{3} = \frac{6}{3}\newlineS=2S = 2\newlineSo, the price for a case of soda is $2\$2.
  5. Solving for J: Now that we know the price of a case of soda, we can substitute S=2S = 2 into one of the original equations to find the price of a case of juice. Let's use the first equation:\newline12J+10S=4412J + 10S = 44\newlineSubstituting S=2S = 2, we get:\newline12J+10(2)=4412J + 10(2) = 44
  6. Final Answer: Now we solve for JJ:12J+20=4412J + 20 = 44Subtract 2020 from both sides:12J=44−2012J = 44 - 2012J=2412J = 24
  7. Final Answer: Now we solve for JJ:12J+20=4412J + 20 = 44Subtract 2020 from both sides:12J=44−2012J = 44 - 2012J=2412J = 24Finally, we divide both sides by 1212 to find JJ:12J12=2412\frac{12J}{12} = \frac{24}{12}J=2J = 2So, the price for a case of juice is also $2\$2.

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