in one day, joe consumes 530 calories by drinking 1 serving of juice, 2 servings of milk, and 1 soda. darius…

Answer:

B. 110

Explanation:

Step1: Define variables

Let $x$ be calories in juice, $y$ be calories in milk, $z$ be calories in soda. We get the system of equations: $x + 2y+z=530$; $2x + y=370$; $3y+z = 510$. In matrix form $A\mathbf{X}=\mathbf{B}$, where $A=\begin{bmatrix}1&2&1\2&1&0\0&3&1\end{bmatrix}$, $\mathbf{X}=\begin{bmatrix}x\y\z\end{bmatrix}$, $\mathbf{B}=\begin{bmatrix}530\370\510\end{bmatrix}$.

Step2: Find the inverse of matrix $A$

First, find the determinant of $A$, $\text{det}(A)=1\times(1\times1 - 0\times3)-2\times(2\times1 - 0\times0)+1\times(2\times3 - 1\times0)$ $=1\times1-2\times2 + 1\times6=1 - 4+6 = 3$. Then find the co - factor matrix, adjugate matrix and inverse matrix $A^{-1}=\frac{1}{3}\begin{bmatrix}1&-1&-1\-2&1&2\6&-3&-3\end{bmatrix}$.

Step3: Solve for $\mathbf{X}$

$\mathbf{X}=A^{-1}\mathbf{B}=\frac{1}{3}\begin{bmatrix}1&-1&-1\-2&1&2\6&-3&-3\end{bmatrix}\begin{bmatrix}530\370\510\end{bmatrix}$ $=\frac{1}{3}\begin{bmatrix}530-370 - 510\-2\times530+370 + 2\times510\6\times530-3\times370-3\times510\end{bmatrix}=\frac{1}{3}\begin{bmatrix}-350\330\0\end{bmatrix}=\begin{bmatrix}\frac{-350}{3}\110\0\end{bmatrix}$. So the calories in 1 serving of milk $y = 110$.