Question

Maya bought a dozen specialty donuts at the bakery for $18. She purchased a mixture of frosted donuts for $2 each, glazed donuts for $1 each, and custard filled donuts for $5 each. If Maya purchased twice as many frosted donuts as custard filled donuts, how many of each type of donut did she buy? need system equations an system in standard form and augmented matrix

Answer 1

x =2/3 y =10 z =-4/3

Gathering the data

a mixture of frosted donuts = 2 x

glazed donuts for $1 y

custard-filled donuts: 5 z

twice the frosted donuts 2 (2x)

custard-filled donuts: 2 (y)

2) Relating the quantities of donuts in the first equation. And in the second, one relating the price.

x+y+z=12

2x +y +5z=18

2x=z

3) Let's write the matrices

`\begin{bmatrix}{1} & {1} & {1} \\ {2} & {1} & {5} \\ {2} & {0} & {-1}\end{bmatrix}\cdot\begin{bmatrix}{x} & {} & \\ {y} & {} & {} \\ {z} & {} & {}\end{bmatrix}=\begin{bmatrix}{12} & {} & {} \\ {18} & {} & {} \\ {0} & {} & {}\end{bmatrix}`

x+y+z=12

2x +y +5z=18

2x=z

x+y+z=12 * -2

2x +y +5z=18 x -1

2x+2y+2z=24

-2x -y -5z=-18

---------------------------

y -3z = 6

y -3z = 6

2x -z =0

x+y+z=12 x (-2)

2x -z =0

-2x -2y-2z = -24

------------------------

-3z +2y=-24

y -3z = 6 x 2

-3z +2y=-24

-6z-2y =12

------------------

-9z = 12

z=-4/3 2x = z 2x -(4/3)=0 x =2/3 y =10