Question

The cost of two groups going to an amusement park is shown in the table.

Group	Total Cost
$4\mathrm{adults}$,$2\mathrm{children}$	$\mathrm{\backslash (}184$
localid="1647549424859" $4\mathrm{adults}$,localid="1647549450994" $3\mathrm{children}$	$\mathrm{\backslash )}200$

a. Define variables to represent the cost of an adult ticket and the cost of a child ticket.

b. Write a system of equations to find the cost of an adult ticket and the cost of a child ticket.

c. Solve the systems of equations, and explain what the solution means.

d. How much will a group of 3 adults and 5 children be charged for admission?

Short answer

a. The cost of an adult ticket be $\mathrm{\$}38$and the cost of a child ticket be $\mathrm{\$}16$.

b. The system of equations is $4x+2y=184$and $4x+3y=200$.

c. The solution of the system of equations is $\left(38,16\right)$.

d. The total amount that will be charged for a group of $3\mathrm{adults}$and $5\mathrm{children}$is $\mathrm{\$}194$.

Step-by-step solution

Step 1. Define the concept.

Two equations of the form $a_{1}x+b_{1}y=c_1$ and$a_{2}x+b_{2}y=c_2$ form a system of equations. The ordered pair that is a solution of both equations is the solution of the system. A system can have one solution, an infinite number of solutions, or no solution.

Step 2. Assign the variables.

The cost of two groups is given which ticket costs of an adult and children.

Let the cost of an adult ticket be x and the cost of a child ticket be y.

Step 3. Create the table to represent the assigned variables.

a. The table that represents the information of the variables is:

adult	children	Cost (in dollars)
4	2	184
4	3	200

b. The table that represents the information of the variables is:

adult	children	Cost (in dollars)
4	2	184
4	3	200

In group-1, the total cost of tickets of 4 adults and 2 children is $\mathrm{\$}184$. So the cost of 4 adult tickets will be $\mathrm{\$}4x$and of 2 child tickets will be $\mathrm{\$}2y$.Therefore, the equation that connects the variable and given quantities is

$4x+2y=184$ … (1)

In group-2, the total cost of tickets for 4 adults and 3 children is $200. So the cost of 4 adult tickets will be $\mathrm{\$}4x$and of 3 child tickets will be $\mathrm{\$}3y$.Therefore, the equation that connects the variable and given quantities is

$4x+3y=200$ … (1)

c. The elimination method is the method in which the addition or subtraction method is used to get an equation in one variable.

If the coefficients are opposite, use the addition and when the coefficients are the same, use the subtraction.

The given system of equation is:

$4x+2y=184$ … (1)

$4x+3y=200$ … (2)

Note that both the equations (1) and (2) have 4x in the same signs, therefore, variable x can be eliminated if both equations are subtracted.

localid="1647547807355" $$\begin{gathered} 4x+2y=184 \\ \underset{¯}{\left(-\right)4x+3y=200} \\ -y=-16\text{Dividebothsidesby-1} \\ y=16 \end{gathered}$$

Substitute 16 for y into the equation (1) and solve for x.

localid="1647547920490" $$\begin{gathered} 4x+2y=184 \\ 4x+2\left(16\right)-32=184-32 \\ 4x=152 \\ x=38 \end{gathered}$$

The solution is $\left(38,16\right)$.

d. The table that represents the information of the variables is:

adult	children	Cost (in dollars)
4	2	184
4	3	200

$\mathrm{\$}184$

In group-1, the total cost of tickets for 4 adults and 2 children is $184. So the cost of 4 adult tickets will be $\mathrm{\$}4x$and of 2 child tickets will be $\mathrm{\$}2y$.Therefore, the equation that connects the variable and given quantities is

$4x+2y=184$ … (1)

In group-2, the total cost of tickets for 4 adults and 3 children is $200. So the cost of 4 adult tickets will be $\mathrm{\$}4x$and of 3 child tickets will be $\mathrm{\$}3y$.Therefore, the equation that connects the variable and given quantities is

$4x+3y=200$ … (1)

After solving the system of equations using the elimination method, the solution is that the cost of 1 adult ticket is $38 and the cost of 1 child ticket is $16.

The cost of 1 adult ticket is $\mathrm{\$}38$and cost of 1 child ticket is $\mathrm{\$}16$.

The amount paid for the ticket of 3 adults is:

localid="1647548471879" $\begin{array}{c}{c}_1=3\times 38\\ =114\end{array}$

The amount paid for 5 child tickets:

localid="1647548463644" $$\begin{gathered} c_2=5\times 16 \\ =80 \end{gathered}$$

Therefore, the total amount that will be charged for a group of 3 adults and 5 children is:

localid="1647548515116" $\begin{array}{c}{c}_1+c_2=114+80\\ =\mathrm{\$}194\end{array}$

Thus, the total amount that will be charged for a group of 3 adults and 5 children is $\mathrm{\$}194$.