company sell? let $x_1$ = the number of widgets sold and $x_2$ = the number of gizmos sold. write an…

Explanation:

Step1: Recall matrix - solving formula

For a system of linear equations $AX = B$, where $A=\begin{bmatrix}a_{11}&a_{12}\a_{21}&a_{22}\end{bmatrix}$, $X=\begin{bmatrix}x_1\x_2\end{bmatrix}$ and $B=\begin{bmatrix}b_1\b_2\end{bmatrix}$, the solution is $X = A^{-1}B$. Here, $A=\begin{bmatrix}1&1\35&22\end{bmatrix}$, $B=\begin{bmatrix}350\9000\end{bmatrix}$.

Step2: Calculate the inverse of matrix $A$

The determinant of $A$, $\text{det}(A)=1\times22 - 1\times35=22 - 35=- 13$. The inverse of $A$, $A^{-1}=\frac{1}{\text{det}(A)}\begin{bmatrix}22&-1\-35&1\end{bmatrix}=-\frac{1}{13}\begin{bmatrix}22&-1\-35&1\end{bmatrix}=\begin{bmatrix}-\frac{22}{13}&\frac{1}{13}\\frac{35}{13}&-\frac{1}{13}\end{bmatrix}$.

Step3: Calculate $X = A^{-1}B$

$X=\begin{bmatrix}-\frac{22}{13}&\frac{1}{13}\\frac{35}{13}&-\frac{1}{13}\end{bmatrix}\begin{bmatrix}350\9000\end{bmatrix}=\begin{bmatrix}-\frac{22}{13}\times350+\frac{1}{13}\times9000\\frac{35}{13}\times350-\frac{1}{13}\times9000\end{bmatrix}=\begin{bmatrix}-\frac{7700}{13}+\frac{9000}{13}\\frac{12250}{13}-\frac{9000}{13}\end{bmatrix}=\begin{bmatrix}\frac{-7700 + 9000}{13}\\frac{12250-9000}{13}\end{bmatrix}=\begin{bmatrix}\frac{1300}{13}\\frac{3250}{13}\end{bmatrix}=\begin{bmatrix}100\250\end{bmatrix}$

Answer:

Widgets sold: $100$; gizmos sold: $250$