a chemist has a container of 12% peroxide solution and a container of 20% peroxide solution. she needs to…

a chemist has a container of 12% peroxide solution and a container of 20% peroxide solution. she needs to mix the two solutions to create 100 ml of a 14% solution. if x represents the amount of 12% peroxide solution and y represents the amount of 20% peroxide solution she needs, which matrix equation can determine the amount of each solution she needs to make the 14% solution?\n \begin{bmatrix}0.12&0.2\\1&1end{bmatrix}\begin{bmatrix}x\\yend{bmatrix}=\begin{bmatrix}0.14\\100end{bmatrix}\n \begin{bmatrix}0.12&0.2\\1&1end{bmatrix}\begin{bmatrix}x\\yend{bmatrix}=\begin{bmatrix}14\\100end{bmatrix}\n \begin{bmatrix}1&1\\0.12&0.2end{bmatrix}\begin{bmatrix}x\\yend{bmatrix}=\begin{bmatrix}14\\100end{bmatrix}\n \begin{bmatrix}1&1\\0.12&0.2end{bmatrix}\begin{bmatrix}x\\yend{bmatrix}=\begin{bmatrix}0.14\\100end{bmatrix}

a chemist has a container of 12% peroxide solution and a container of 20% peroxide solution. she needs to mix the two solutions to create 100 ml of a 14% solution. if x represents the amount of 12% peroxide solution and y represents the amount of 20% peroxide solution she needs, which matrix equation can determine the amount of each solution she needs to make the 14% solution?\n \begin{bmatrix}0.12&0.2\\1&1end{bmatrix}\begin{bmatrix}x\\yend{bmatrix}=\begin{bmatrix}0.14\\100end{bmatrix}\n \begin{bmatrix}0.12&0.2\\1&1end{bmatrix}\begin{bmatrix}x\\yend{bmatrix}=\begin{bmatrix}14\\100end{bmatrix}\n \begin{bmatrix}1&1\\0.12&0.2end{bmatrix}\begin{bmatrix}x\\yend{bmatrix}=\begin{bmatrix}14\\100end{bmatrix}\n \begin{bmatrix}1&1\\0.12&0.2end{bmatrix}\begin{bmatrix}x\\yend{bmatrix}=\begin{bmatrix}0.14\\100end{bmatrix}

Answer

Explanation:

Step1: Set up the amount - relationship equation

The total volume of the final solution is 100 mL. So, $x + y=100$. In matrix - form, the coefficients of $x$ and $y$ for this equation are $\begin{bmatrix}1&1\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=100$.

Step2: Set up the concentration - relationship equation

The amount of peroxide in the 12% solution is $0.12x$, the amount of peroxide in the 20% solution is $0.2y$, and the amount of peroxide in the 14% solution of 100 mL is $0.14\times100 = 14$. So, $0.12x+0.2y = 14$. In matrix - form, the coefficients are $\begin{bmatrix}0.12&0.2\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=14$. Combining these two equations into a matrix equation gives $\begin{bmatrix}0.12&0.2\1&1\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}14\100\end{bmatrix}$.

Answer:

$\begin{bmatrix}0.12&0.2\1&1\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}14\100\end{bmatrix}$ (corresponding to the second option in the multiple - choice list)