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 $16$ packages of coffee and $13$ packages of tea, for which customers paid a total of $\$235$ . The day before, $33$ packages of coffee and $13$ packages of tea was sold, which brought in a total of $\$388$ . 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

16

packages of coffee and

13

packages of tea, for which customers paid a total of

\$235

. The day before,

33

packages of coffee and

13

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

\$388

. How much does each package cost?

\newline

Per package, coffee costs

\$\_

and tea costs

\$\_

Equations Setup: Let's denote the cost of a package of coffee as $c$ dollars and the cost of a package of tea as $t$ dollars. We can write two equations based on the information given: $\newline$ $1$ . For the first day: $16c + 13t = 235$ $\newline$ $2$ . For the second day: $33c + 13t = 388$ $\newline$ These two equations form our system of equations.
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: $\newline$ $(33c + 13t) - (16c + 13t) = 388 - 235$ $\newline$ This simplifies to: $\newline$ $17c = 153$
Solving for Coffee Cost: Now we can solve for $c$ by dividing both sides of the equation by $17$ : $\newline$ $c = \frac{153}{17}$ $\newline$ $c = 9$ $\newline$ So, a package of coffee costs $9$.
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$$ 144 + 13t = 235 $
Solving for Tea Cost: Now, we subtract $144$ from both sides to solve for $ t $:$\newline$$ 13t = 235 - 144 $$\newline$$ 13t = 91 $
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 $7$ .