一家主题公园根据顾客的年龄实行不同的门票价格。年龄类别分为老年人（60岁及以上）、成人（13至59岁）和儿童（4至12岁）。四个旅行团在同一天游览了该主题公园。下表给出了其中三个旅行团在各个类别中的人数，以及每个旅行团的门票总费用。

旅行团	老年人	成人	儿童	总费用
$A$	2	19	9	$1982
$B$	0	10	3	$908
$C$	1	7	4	$778

求旅行团 $D$ 的门票总费用，该旅行团由四名老年人、五名成人和一名儿童组成。

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.
$\text{Total cost} = 4x + 5y + z$
$\text{Total cost} = 4(36) + 5(74) + 1(56)$
$\text{Total cost} = 144 + 370 + 56 = 570$