Matt bought 3 pounds of oranges and 2 pounds of apples and paid $\$ 4.14$, before tax. Andy bought 4 pounds of oranges and 3 pounds of apples and paid $\$ 5.83$, before tax. Use this information to set up a matrix equation of the form $\mathrm { AX } = \mathrm { B }$, which can be solved to determine the price per pound for oranges and apples. Solve this matrix equation to find the price per pound of oranges.
Use the fact that for $A = \left[ \begin{array} { l l } 3 & 2 \ 4 & 3 \end{array} \right] , A ^ { - 1 } = \left[ \begin{array} { r r } 3 & - 2 \ - 4 & 3 \end{array} \right]$.
A) $0.72 per pound
B) $0.76 per pound
C) $0.79 per pound
D) $0.74 per pound

Question

Quizplus · Accepted Answer

The matrix equation is $AX = B$, where $A = \left[ \begin{array} { l l } 3 & 2 \ 4 & 3 \end{array} \right]$, $X$ is the matrix of unknown prices per pound for oranges and apples, and $B = \left[ \begin{array} { l } 4.14 \ 5.83 \end{array} \right]$. To find $X$, we multiply both sides by $A^{-1}$, so $X = A^{-1}B$. Given $A^{-1} = \left[ \begin{array} { r r } 3 & -2 \ -4 & 3 \end{array} \right]$, we calculate $X = A^{-1}B = \left[ \begin{array} { r r } 3 & -2 \ -4 & 3 \end{array} \right] \left[ \begin{array} { l } 4.14 \ 5.83 \end{array} \right]$. This results in $X = \left[ \begin{array} { l } 0.76 \ ? \end{array} \right]$, where 0.76 is the price per pound of oranges.

Matt Bought 3 Pounds of Oranges and 2 Pounds of Apples