Question

a chemist mixes different concentrations of perfume using three scents the company already produces. scents a, b, and c have different concentrations of rose oil. the results of three different trials are listed in the table. the result is the percent of rose oil by volume for each trial.

a	b	c	result
4 oz	2 oz	4 oz	27%
1 oz	7 oz	2 oz	23.5%

how many total ounces of perfume are in the result in each trial (each row)? 10 oz.

let (x_1,x_2,) and (x_3) represent the concentration of rose oil in each sample. fill in the matrix equation:
\begin{bmatrix} & & \\ & & \\ & & end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 end{bmatrix} = \begin{bmatrix} 2.2 \\ 2.7 \\ 2.35 end{bmatrix}

Explanation:

Step1: Recall matrix - equation formation for mixture problems

For a mixture problem, if we have amounts of different substances and their resulting concentrations, the left - hand side of the matrix equation is a matrix of the amounts of each substance in each trial and the right - hand side is a vector of the actual amounts of the component (in this case, rose oil) in each trial.

Step2: Identify the amounts of scents A, B, and C in each trial

In the first row (trial 1), we have 2 oz of A, 4 oz of B, and 4 oz of C. In the second row (trial 2), we have 4 oz of A, 2 oz of B, and 4 oz of C. In the third row (trial 3), we have 1 oz of A, 7 oz of B, and 2 oz of C.

Step3: Write the matrix equation

The matrix equation is

$$\begin{bmatrix}2&4&4\\4&2&4\\1&7&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}2.2\\2.7\\2.35\end{bmatrix}$$

Answer:

$$\begin{bmatrix}2&4&4\\4&2&4\\1&7&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}2.2\\2.7\\2.35\end{bmatrix}$$