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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 1616 packages of coffee and 1313 packages of tea, for which customers paid a total of $235\$235. The day before, 3333 packages of coffee and 1313 packages of tea was sold, which brought in a total of $388\$388. 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 1616 packages of coffee and 1313 packages of tea, for which customers paid a total of $235\$235. The day before, 3333 packages of coffee and 1313 packages of tea was sold, which brought in a total of $388\$388. How much does each package cost?\newlinePer package, coffee costs $_\$\_ and tea costs $_\$\_.
  1. Equations Setup: Let's denote the cost of a package of coffee as c c dollars and the cost of a package of tea as t t dollars. We can write two equations based on the information given:\newline11. For the first day: 16c+13t=235 16c + 13t = 235 \newline22. For the second day: 33c+13t=388 33c + 13t = 388 \newlineThese two equations form our system of equations.
  2. Elimination Method: To use elimination, we want to eliminate one of the variables. We can do this by subtracting the first equation from the second equation:\newline(33c+13t)−(16c+13t)=388−235 (33c + 13t) - (16c + 13t) = 388 - 235 \newlineThis simplifies to:\newline17c=153 17c = 153
  3. Solving for Coffee Cost: Now we can solve for c c by dividing both sides of the equation by 1717:\newlinec=15317 c = \frac{153}{17} \newlinec=9 c = 9 \newlineSo, a package of coffee costs \(9\).
  4. Substitution for Tea Cost: With the value of \( c \) known, we can substitute it back into one of the original equations to solve for \( t \). Let's use the first equation:\(\newline\)\( 16(9) + 13t = 235 \)\(\newline\)\( 144 + 13t = 235 \)
  5. Solving for Tea Cost: Now, we subtract \(144\) from both sides to solve for \( t \):\(\newline\)\( 13t = 235 - 144 \)\(\newline\)\( 13t = 91 \)
  6. Solving for Tea Cost: Now, we subtract \(144\) from both sides to solve for \( t \):\(\newline\)\( 13t = 235 - 144 \)\(\newline\)\( 13t = 91 \)Finally, we divide both sides by \(13\) to find the value of \( t \):\(\newline\)\( t = \frac{91}{13} \)\(\newline\)\( t = 7 \)\(\newline\)So, a package of tea costs 77.

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