Use a system of linear equations in three variables to solve the following problem.
A certain brand of razor blade comes in packages of 6,12 , and 24 blades, costing $\mathrm{\$}2$, $\mathrm{\$}3$, and $\mathrm{\$}4$ per package, respectively. A store sold 16 packages containing a total of 210 razor blades and took in $\mathrm{\$}46$. How many packages of each type were sold?
Write a system of linear equations using the given information. Choose correct answer below.
A. $\{\begin{array}{l}6x+12y+24z=16\\ 2x+3y+4z=210\\ x+y+z=46\end{array}$
$\{\begin{array}{l}6x+12y+24z=210\\ 2x+3y+4z=46\\ x+y+z=16\end{array}$
C. $\{\begin{array}{l}6x+12y+24z=46\\ 2x+3y+4z=16\\ x+y+z=210\end{array}$
D. $\{\begin{array}{l}6x+12y+24z=46\\ 2x+3y+4z=210\\ x+y+z=16\end{array}$
There were packages of 6 blades, packages of 12 blades, and packages of 24 blades sold.

Step 1 ：The problem involves three variables, x, y, and z, which represent the number of packages of 6, 12, and 24 blades sold, respectively. We are given three pieces of information: the total number of packages sold (16), the total number of blades sold (210), and the total revenue ($46). We can use these to form a system of linear equations.

Step 2 ：The total number of packages sold is the sum of the number of each type of package sold, so we have the equation $x+y+z=16$.

Step 3 ：The total number of blades sold is the sum of the number of blades in each type of package sold, so we have the equation $6x+12y+24z=210$.

Step 4 ：The total revenue is the sum of the cost of each type of package sold, so we have the equation $2x+3y+4z=46$.

Step 5 ：So, the correct system of linear equations is: $$\begin{array}{rl}x+y+z& =16\\ 6x+12y+24z& =210\\ 2x+3y+4z& =46\end{array}$$

Step 6 ：This corresponds to option B.

Step 7 ：Final Answer: The correct system of linear equations is $\boxed{{\displaystyle B}}$.