Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ The drama club is selling gift baskets to raise money for new costumes. During the fall play, they sold a combined $9$ regular gift baskets and $3$ deluxe gift baskets, earning a total of $\$297$ . During the spring musical, they sold $1$ deluxe gift basket, earning a total of $\$45$ . How much are they charging for the different sized gift baskets? $\newline$ The drama club is charging $\$$ $\_\_\_\_$ for a regular gift basket and $\$$ $\_\_\_\_$ for a deluxe gift basket.

Full solution

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

\newline

The drama club is selling gift baskets to raise money for new costumes. During the fall play, they sold a combined

9

regular gift baskets and

3

deluxe gift baskets, earning a total of

\$297

. During the spring musical, they sold

1

deluxe gift basket, earning a total of

\$45

. How much are they charging for the different sized gift baskets?

\newline

The drama club is charging

\$

\_\_\_\_

for a regular gift basket and

\$

\_\_\_\_

for a deluxe gift basket.

Write Equations: Let $r$ be the price of a regular gift basket and $d$ be the price of a deluxe gift basket. We can write two equations based on the information given: $\newline$ $1$ . For the fall play: $9r + 3d = 297$ $\newline$ $2$ . For the spring musical: $d = 45$
Create Augmented Matrix: To solve using an augmented matrix, we first write the coefficients of $r$ and $d$ in matrix form. The augmented matrix is: $\newline$ \begin{array}{cc|c} 9 & 3 & 297 \ 0 & 1 & 45 \end{array}
Substitute and Solve: Since the second row already shows that $d = 45$ , we can use this value to find $r$ . We substitute $d$ into the first equation: $\newline$ $9r + 3(45) = 297$
Find $r$ : Now we solve for $r$ : $\newline$ $9r + 135 = 297$ $\newline$ $9r = 297 - 135$ $\newline$ $9r = 162$
Final Answer: Divide both sides by $9$ to find $r$ : $r = \frac{162}{9}$ $r = 18$