Chapter 9.7, Problem 22IP

To determine

To represent : the given situation by a system of equations and solve it.

To interpret the solution of the system of equations.

Answer to Problem 22IP

The system of equations representing the given situation:

  $\begin{array}{l}y=2x\\ 6x+8y=110\end{array}$

The solution of the system of equations is $\text{(5},10)$ .

From the solution of the system of equations, it can be concluded that the owners will use $5$ six-seat tables and $\text{10}$ eight-seat tables to seat a $110$ people.

Explanation of Solution

Given information :

The owners of a catering business need to seat $110$ people for a banquet. Their tables seat $6$ or $\text{8}$ people. They want to use twice as many tables that seat $6$ as tables that seat $\text{8}$ . Each table will seat its maximum number of people.

Calculation :

As per problem,

Let the number of $6$ -seat tablebe $x$ and that of $\text{8}$ -seat tablesbe $y$ .

Twice as many tables that seat $6$ are to be used as tables that seat $\text{8}$ .

Total number of people to be seatedis $110$ .Each table will seat its maximum number of people.

Therefore, the system of equations for the given situation:

  $\begin{array}{l}y=2x\\ 6x+8y=110\end{array}$

Find the value of $x$ by substituting $2x$ for $y$ in the second equation.

  $\begin{array}{l}\text{ 6}x+8y=110\text{ [Write the second equation]}\\ \text{6}x+8(2x)=110\text{ [Substitute }y=2x\text{]}\\ 6x+16x=110\text{ [Simplify}]\\ \text{ 22}x=110\text{ [Combine like terms}]\\ x=5\text{ [Divide both sides by 22}]\end{array}$

Substitute $5$ for $x$ in either equation to find the value of $y$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}y=2x\text{ [Write the first equation]}\\ y=2(5)\text{ [}x=5\text{]}\\ y=10\text{ [Simplify}]\end{array}$

The solution of this system of equations is $\text{(5},10)$ .

Thus, the owners will use $5$ six-seat tables and $\text{10}$ eight-seat tables to seat $110$ people.