Question

Suppose Roberto, Travis and Deandre go to adonut shop.Roberto buys 6 amaretto pumpkin, 6 black forestand 8 cotton candy donuts and ends up spending$48.70Travis gets 5 of the amaretto pumpkin, 6 blackforest and 6 cotton candy donuts for a total costof $41.33.Lastly, Deandre spends $31.67 on 4 amarettopumpkin, 4 black forest, and 5 cotton candy donuts.Write this scenario as a system of equations, then solve that system using an augmented matrixand RREF on your calculator before answering the question below.How much does a single black forest donut cost? Round your answer to 2 decimal places.

Answer 1

Given:

Let x be the cost of an amaretto pumpkin.

Let y be the cost of a black forest donut.

Let z be the cost of a cotton candy donut

To find:

The cost of a single black forest donut:

Step-by-step explanation:

Let us frame the system of equations as follows.

`\begin{gathered} 6x+6y+8z=48.7 \\ 5x+6y+6z=42.33_{} \\ 4x+4y+5z=31.67 \end{gathered}`

Next, we write the augmented matrix for the system,

`\begin{bmatrix}{6} & {6} & {8} & {48.7} \\ {5} & {6} & {6} & {42.33} \\ {4} & {4} & {5} & {31.67} \\ {} & {} & {} & \end{bmatrix}`

Solving the matrix using RREF calculator,

The reduced row echelon form is,

`\begin{bmatrix}{1} & {0} & {0} & {2.59} \\ {0} & {1} & {0} & {2.34} \\ {0} & {0} & {1} & {2.39} \\ & {} & {} & \end{bmatrix}`

Hence, the values are,

`\begin{gathered} x=2.59 \\ y=2.34 \\ z=2.39 \end{gathered}`

Thus, the cost of a single black forest donut is $2.34.