Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Angie wanted to stock up on drinks for an upcoming party. First, she spent $\$44$ on $12$ cases of juice and $10$ cases of soda, which is all the store had in stock. A few days later, she returned to the store and purchased an additional $12$ cases of juice and $13$ cases of soda, spending a total of $\$50$ . What is the price of each drink? $\newline$ The price for a case of juice is $\$\_\_\_\_$ , and the price for a case of soda is $\$\_\_\_\_\_\_\_\_\_$ .

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Grade 8

Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Angie wanted to stock up on drinks for an upcoming party. First, she spent

\$44

12

cases of juice and

10

cases of soda, which is all the store had in stock. A few days later, she returned to the store and purchased an additional

12

cases of juice and

13

cases of soda, spending a total of

\$50

. What is the price of each drink?

\newline

The price for a case of juice is

\$\_\_\_\_

, and the price for a case of soda is

\$\_\_\_\_\_\_\_\_\_

Forming Equations: Let's denote the price of a case of juice as $J$ dollars and the price of a case of soda as $S$ dollars. We can write two equations based on the information given: $\newline$ First purchase: $12J + 10S = 44$ $\newline$ Second purchase: $12J + 13S = 50$ $\newline$ We will use these two equations to form a system of equations that we can solve using the elimination method.
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 $J$ is the same in both equations, we can subtract the first equation from the second to eliminate $J$ . $\newline$ Subtracting the first equation from the second gives us: $\newline$ $(12J + 13S) - (12J + 10S) = 50 - 44$
Solving for S: Performing the subtraction, we get: $\newline$ $12J + 13S - 12J - 10S = 50 - 44$ $\newline$ This simplifies to: $\newline$ $3S = 6$
Substitute S: Now we can solve for S by dividing both sides of the equation by $3$ : $\newline$ $\frac{3S}{3} = \frac{6}{3}$ $\newline$ $S = 2$ $\newline$ So, the price for a case of soda is $\$2$ .
Solving for J: Now that we know the price of a case of soda, we can substitute $S = 2$ into one of the original equations to find the price of a case of juice. Let's use the first equation: $\newline$ $12J + 10S = 44$ $\newline$ Substituting $S = 2$ , we get: $\newline$ $12J + 10(2) = 44$
Final Answer: Now we solve for $J$ : $12J + 20 = 44$ Subtract $20$ from both sides: $12J = 44 - 20$ $12J = 24$
Final Answer: Now we solve for $J$ : $12J + 20 = 44$ Subtract $20$ from both sides: $12J = 44 - 20$ $12J = 24$ Finally, we divide both sides by $12$ to find $J$ : $\frac{12J}{12} = \frac{24}{12}$ $J = 2$ So, the price for a case of juice is also $\$2$ .