Respuesta :
Answer:
$3.5
Explanation:
Let the cost of 1 kg of apples = x
Let the cost of 1 kg of bananas =y
Claire bought 5 kg of apples and 2 kg of bananas and paid altogether $22.
[tex]5x+2y=22[/tex]Dale bought 4 kg of apples and 6 kg of bananas and paid altogether $33.
[tex]4x+6y=33[/tex]We set up the system of linear equations as a matrix below:
[tex]\begin{bmatrix}{5} & {2} & \\ {4} & {6} & {}{}\end{bmatrix}\begin{bmatrix}{x} & \\ {y} & {}{}\end{bmatrix}=\begin{bmatrix}{22} & \\ {33} & {}{}\end{bmatrix}[/tex]We then solve for the variables x and y as follows.
[tex]\begin{gathered} \begin{bmatrix}{x} & \\ {y} & {}{}\end{bmatrix}=\begin{bmatrix}{5} & {2} & \\ {4} & {6} & {}{}\end{bmatrix}^{-1}\begin{bmatrix}{22} & \\ {33} & {}{}\end{bmatrix} \\ =\frac{1}{30-8}\begin{bmatrix}{6} & {-2} & \\ {-4} & {5} & {}{}\end{bmatrix}\begin{bmatrix}{22} & \\ {33} & {}{}\end{bmatrix} \\ =\frac{1}{22}\begin{bmatrix}{6} & {-2} & \\ {-4} & {5} & {}{}\end{bmatrix}\begin{bmatrix}{22} & \\ {33} & {}{}\end{bmatrix} \end{gathered}[/tex]We proceed to simplify further.
[tex]\begin{gathered} =\begin{bmatrix}{\frac{6}{22}} & {-\frac{2}{22}} & \\ {-\frac{4}{22}} & {\frac{5}{22}} & {}{}\end{bmatrix}\begin{bmatrix}{22} & \\ {33} & {}{}\end{bmatrix} \\ =\begin{bmatrix}{\frac{6}{22}\times22-\frac{2}{22}\times33} & {} & \\ {\frac{-4}{22}\times22+\frac{5}{22}\times33} & & {}{}\end{bmatrix} \\ =\begin{bmatrix}{6-3} & {} & \\ {-4+7.5} & & {}{}\end{bmatrix} \\ =\begin{bmatrix}{3} & {} & \\ {3.5} & & {}{}\end{bmatrix} \end{gathered}[/tex]Therefore:
x=3 and y=3.5.
The cost of 1 kg of bananas is $3.5.
