A farmer has 100 acres of land on which tomatoes, peas and carrots can be planted. The labour cost, materials cost and gross income for one acre of these crops are summarised as follows.

	Labour (hr)	Materials Cost ($)	Gross Income ($)
Tomatoes	30	50	500
Peas	45	30	225
Carrots	50	40	325

The farmer has 4000 hours of labour and $4250 of capital to invest. Find the distribution of land area for each crop and the total gross income for the farmer.

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 $= 500(45) + 225(20) + 325(35)$
$= 22500 + 4500 + 11375$
$= \char36{}38375$