Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ A coffee shop is having a sale on prepackaged coffee and tea. Yesterday they sold $10$ packages of coffee and $31$ packages of tea, for which customers paid a total of $\$204$ . The day before, $10$ packages of coffee and $47$ packages of tea was sold, which brought in a total of $\$268$ . How much does each package cost? $\newline$ Per package, coffee costs $\$$ _ and tea costs $\$$ _.

Full solution

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

\newline

A coffee shop is having a sale on prepackaged coffee and tea. Yesterday they sold

10

packages of coffee and

31

packages of tea, for which customers paid a total of

\$204

. The day before,

10

packages of coffee and

47

packages of tea was sold, which brought in a total of

\$268

. How much does each package cost?

\newline

Per package, coffee costs

\$

_____ and tea costs

\$

_____.

Define Prices: Let's denote the price of each package of coffee as $c$ and the price of each package of tea as $t$ . From the first day's sales, we have the equation $10c + 31t = 204$ .
Sales Equations: From the second day's sales, we have the equation $10c + 47t = 268$ .
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 $c$ .
Subtract Equations: Subtracting the first equation from the second gives us $10c + 47t - (10c + 31t) = 268 - 204$ , which simplifies to $16t = 64$ .
Find Tea Price: Dividing both sides of the equation $16t = 64$ by $16$ gives us $t = 4$ . This means that each package of tea costs $\$4$ .
Substitute Tea Price: Now that we know the price of tea, we can substitute $t = 4$ into the first equation to find the price of coffee. Substituting gives us $10c + 31(4) = 204$ .
Solve for Coffee Price: Solving for $c$ gives us $10c + 124 = 204$ . Subtracting $124$ from both sides gives us $10c = 80$ .
Solve for Coffee Price: Solving for $c$ gives us $10c + 124 = 204$ . Subtracting $124$ from both sides gives us $10c = 80$ . Dividing both sides by $10$ gives us $c = 8$ . This means that each package of coffee costs $\$8$ .