一位农民拥有 100 英亩土地，可以用来种植西红柿、豌豆和胡萝卜。种植每英亩这些作物所需的人工（小时）、材料成本（$）和总收入（$）汇总如下。

	人工 (小时)	材料成本 ($)	总收入 ($)
西红柿	30	50	500
豌豆	45	30	225
胡萝卜	50	40	325

该农民有 4000 小时的人工和 $4250 的资金可供投入。求每种作物的土地面积分配以及该农民的总收入。

1. Define variables and set up the system of linear equations representing the constraints.

Let $x$, $y$, and $z$ be the number of acres planted with tomatoes, peas, and carrots respectively.
Based on the total land area, labour hours, and materials cost:
$x + y + z = 100$ --- (1)
$30x + 45y + 50z = 4000$ --- (2)
$50x + 30y + 40z = 4250$ --- (3)

2. Simplify the equations and eliminate one variable.

From (1), we can express $z$ in terms of $x$ and $y$:
$z = 100 - x - y$ --- (4)

Substitute (4) into (2):
$30x + 45y + 50(100 - x - y) = 4000$
$30x + 45y + 5000 - 50x - 50y = 4000$
$-20x - 5y = -1000$
$4x + y = 200 \implies y = 200 - 4x$ --- (5)

Substitute (4) into (3):
$50x + 30y + 40(100 - x - y) = 4250$
$50x + 30y + 4000 - 40x - 40y = 4250$
$10x - 10y = 250$
$x - y = 25$ --- (6)

3. Solve for x, y, and z.

Substitute (5) into (6):
$x - (200 - 4x) = 25$
$5x - 200 = 25$
$5x = 225 \implies x = 45$

Substitute $x = 45$ into (5):
$y = 200 - 4(45) = 20$

Substitute $x = 45$ and $y = 20$ into (4):
$z = 100 - 45 - 20 = 35$

(Note: This system can also be solved directly using the system solver app on a graphing calculator.)

The distribution is 45 acres of tomatoes, 20 acres of peas, and 35 acres of carrots.

4. Calculate the total gross income.

Total Gross Income $= 500x + 225y + 325z$
Total Gross Income $= 500(45) + 225(20) + 325(35)$
$= 22500 + 4500 + 11375$
$= \char36{}38375$