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 $\$ 2$, $\$ 3$, and $\$ 4$ per package, respectively. A store sold 16 packages containing a total of 210 razor blades and took in $\$ 46$. How many packages of each type were sold?
Write a system of linear equations using the given information. Choose correct answer below.
A. $\left\{\begin{array}{l}6 x+12 y+24 z=16 \\ 2 x+3 y+4 z=210 \\ x+y+z=46\end{array}\right.$
$\left\{\begin{array}{l}6 x+12 y+24 z=210 \\ 2 x+3 y+4 z=46 \\ x+y+z=16\end{array}\right.$
C. $\left\{\begin{array}{l}6 x+12 y+24 z=46 \\ 2 x+3 y+4 z=16 \\ x+y+z=210\end{array}\right.$
D. $\left\{\begin{array}{l}6 x+12 y+24 z=46 \\ 2 x+3 y+4 z=210 \\ x+y+z=16\end{array}\right.$
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: \[\begin{align*} x + y + z &= 16 \\ 6x + 12y + 24z &= 210 \\ 2x + 3y + 4z &= 46 \end{align*}\]

Step 6 ：This corresponds to option B.

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