Question

use a system of linear equations to solve the following problem. a new restaurant is to contain two - seat tables and four - seat tables. fire codes limit the restaurants maximum occupancy to 64 customers. if the owners have hired enough servers to handle 19 tables of customers, how many of each kind of table should they purchase? write a system of linear equations using the given information. choose correct answer below. a. $\begin{cases}x - y = 64\\2x - 4y = 19end{cases}$ b. $\begin{cases}2x + 4y = 64\\x + y = 19end{cases}$ c. $\begin{cases}x + y = 64\\2x + 4y = 19end{cases}$ d. $\begin{cases}2x - 4y = 64\\x - y = 19end{cases}$ they should purchase two - seat tables and four - seat tables.

Explanation:

Step1: Identify the correct system of equations

Let $x$ be the number of two - seat tables and $y$ be the number of four - seat tables. The total number of tables is 19, so $x + y=19$. The total number of customers is 64, and since each two - seat table holds 2 customers and each four - seat table holds 4 customers, we have $2x + 4y=64$. So the correct system is

$$\begin{cases}2x + 4y=64\\x + y=19\end{cases}$$

, which is option B.

Step2: Solve the system of equations

From the second equation $x + y=19$, we can express $x$ as $x = 19 - y$.
Substitute $x = 19 - y$ into the first equation $2x+4y = 64$:
$2(19 - y)+4y=64$.
Expand: $38-2y + 4y=64$.
Combine like terms: $38 + 2y=64$.
Subtract 38 from both sides: $2y=64 - 38=26$.
Divide both sides by 2: $y = 13$.

Step3: Find the value of $x$

Substitute $y = 13$ into $x=19 - y$, we get $x=19 - 13 = 6$.

Answer:

They should purchase 6 two - seat tables and 13 four - seat tables.