A theme park sells tickets at different prices according to the age of the customer. The age categories are senior citizen (ages 60 and above), adult (ages 13 to 59) and child (ages 4 to 12). Four tour groups visited the theme park on the same day. The numbers in each category for three of the groups, together with the total cost of the tickets for each of these groups, are given in the following table.

Group	Senior Citizen	Adult	Child	Total cost
$A$	2	19	9	$1982
$B$	0	10	3	$908
$C$	1	7	4	$778

Find the total cost of the tickets for Tour Group $D$, which consists of four senior citizens, five adults and one child.

1. Formulate a system of linear equations based on the given table.

Let $x$, $y$, and $z$ be the price of a senior citizen, adult, and child ticket respectively.
From the table, we can form the following equations:
$2x + 19y + 9z = 1982 \quad \text{--- (1)}$
$10y + 3z = 908 \quad \text{--- (2)}$
$x + 7y + 4z = 778 \quad \text{--- (3)}$

2. Solve the system of linear equations to find the ticket prices.

From (3), we can express $x$ in terms of $y$ and $z$:
$x = 778 - 7y - 4z$

Substitute this into (1):
$2(778 - 7y - 4z) + 19y + 9z = 1982$
$1556 - 14y - 8z + 19y + 9z = 1982$
$5y + z = 426$
$z = 426 - 5y$

Substitute $z = 426 - 5y$ into (2):
$10y + 3(426 - 5y) = 908$
$10y + 1278 - 15y = 908$
$-5y = -370$
$y = 74$

Now, find $z$ and $x$:
$z = 426 - 5(74) = 56$
$x = 778 - 7(74) - 4(56) = 36$
(Note: This system can also be solved directly using a graphing calculator.)

3. Calculate the total cost for Tour Group D.

Group D consists of $4$ senior citizens, $5$ adults, and $1$ child.
$\text{Total cost} = 4x + 5y + z$
$\text{Total cost} = 4(36) + 5(74) + 1(56)$
$\text{Total cost} = 144 + 370 + 56 = 570$