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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 1010 packages of coffee and 3131 packages of tea, for which customers paid a total of $204\$204. The day before, 1010 packages of coffee and 4747 packages of tea was sold, which brought in a total of $268\$268. 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 1010 packages of coffee and 3131 packages of tea, for which customers paid a total of $204\$204. The day before, 1010 packages of coffee and 4747 packages of tea was sold, which brought in a total of $268\$268. How much does each package cost?\newlinePer package, coffee costs $\$_____ and tea costs $\$_____.
  1. Define Prices: Let's denote the price of each package of coffee as cc and the price of each package of tea as tt. From the first day's sales, we have the equation 10c+31t=20410c + 31t = 204.
  2. Sales Equations: From the second day's sales, we have the equation 10c+47t=26810c + 47t = 268.
  3. Elimination Method: We now have a system of two equations. To solve using elimination, we can subtract the first equation from the second to eliminate the variable cc.
  4. Subtract Equations: Subtracting the first equation from the second gives us 10c+47t−(10c+31t)=268−20410c + 47t - (10c + 31t) = 268 - 204, which simplifies to 16t=6416t = 64.
  5. Find Tea Price: Dividing both sides of the equation 16t=6416t = 64 by 1616 gives us t=4t = 4. This means that each package of tea costs $4\$4.
  6. Substitute Tea Price: Now that we know the price of tea, we can substitute t=4t = 4 into the first equation to find the price of coffee. Substituting gives us 10c+31(4)=20410c + 31(4) = 204.
  7. Solve for Coffee Price: Solving for cc gives us 10c+124=20410c + 124 = 204. Subtracting 124124 from both sides gives us 10c=8010c = 80.
  8. Solve for Coffee Price: Solving for cc gives us 10c+124=20410c + 124 = 204. Subtracting 124124 from both sides gives us 10c=8010c = 80. Dividing both sides by 1010 gives us c=8c = 8. This means that each package of coffee costs $8\$8.

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