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 $25$ packages of coffee and $39$ packages of tea, for which customers paid a total of $\$587$ . The day before, $10$ packages of coffee and $39$ packages of tea was sold, which brought in a total of $\$422$ . How much does each package cost? $\newline$ Per package, coffee costs $\$\_\_\_\_$ and tea costs $\$\_\_\_\_$ .

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Algebra 2

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

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

25

packages of coffee and

39

packages of tea, for which customers paid a total of

\$587

. The day before,

10

packages of coffee and

39

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

\$422

. How much does each package cost?

\newline

Per package, coffee costs

\$\_\_\_\_

and tea costs

\$\_\_\_\_

Equation $1$ : 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 $25c + 39t = 587$ . This represents the total amount earned from selling $25$ packages of coffee and $39$ packages of tea.
Equation $2$ : From the sales of the day before, we have the equation $10c + 39t = 422$ . This represents the total amount earned from selling $10$ packages of coffee and $39$ packages of tea.
Elimination Method: We now have a system of two equations: $\newline$ $1$ . $25c + 39t = 587$ $\newline$ $2$ . $10c + 39t = 422$ $\newline$ We will use elimination to solve this system. To eliminate $t$ , we can subtract the second equation from the first equation.
Subtraction Result: Subtracting the second equation from the first, we get: $\newline$ $(25c + 39t) - (10c + 39t) = 587 - 422$ $\newline$ This simplifies to: $\newline$ $15c = 165$
Coffee Price Calculation: Dividing both sides of the equation $15c = 165$ by $15$ , we find the price of each package of coffee: $\newline$ $c = \frac{165}{15}$ $\newline$ $c = 11$
Tea Price Calculation: Now that we know $c = 11$ , we can substitute this value into one of the original equations to find $t$ . We'll use the second equation: $\newline$ $10c + 39t = 422$ $\newline$ Substituting $c$ with $11$ gives us: $\newline$ $10(11) + 39t = 422$
Tea Price Calculation: Now that we know $c = 11$ , we can substitute this value into one of the original equations to find $t$ . We'll use the second equation: $\newline$ $10c + 39t = 422$ $\newline$ Substituting $c$ with $11$ gives us: $\newline$ $10(11) + 39t = 422$ Solving for $t$ , we get: $\newline$ $110 + 39t = 422$ $\newline$ Subtracting $110$ from both sides gives us: $\newline$ $39t = 422 - 110$ $\newline$ $t$ $0$
Tea Price Calculation: Now that we know $c = 11$ , we can substitute this value into one of the original equations to find $t$ . We'll use the second equation: $\newline$ $10c + 39t = 422$ $\newline$ Substituting $c$ with $11$ gives us: $\newline$ $10(11) + 39t = 422$ Solving for $t$ , we get: $\newline$ $110 + 39t = 422$ $\newline$ Subtracting $110$ from both sides gives us: $\newline$ $39t = 422 - 110$ $\newline$ $t$ $0$ Dividing both sides of the equation $t$ $0$ by $t$ $2$ , we find the price of each package of tea: $\newline$ $t$ $3$ $\newline$ $t$ $4$