If 4 exercise books, 1 mathematical set, and 6 pens cost 22000; 3 exercise books, 3 mathematical sets, and 4 pens cost 25000; and 5 exercise books, 2 mathematical sets, and 3 pens cost $26000, find th

Let $b$ be the cost of one exercise book, $s$ be the cost of one mathematical set, and $p$ be the cost of one pen.
From the given information, we can form the following system of linear equations:

From equation (1), express $s$ in terms of $b$ and $p$:

Substitute (4) into (2):

Substitute (4) into (3):

Divide by 3:

From (6), express $b$ in terms of $p$:

Substitute (7) into (5):

Substitute $p = 1000$ into (7):

Substitute $b = 3000$ and $p = 1000$ into (4):

The cost of each item is:
Book ($b$): 3000 shillings
Set ($s$): 4000 shillings
Pen ($p$): 1000 shillings

Option A is to buy 12 books, 4 sets, and 10 pens.
Cost of Option A $= 12b + 4s + 10p$
Cost of Option A $= 12(3000) + 4(4000) + 10(1000)$
Cost of Option A $= 36000 + 16000 + 10000$
Cost of Option A $= 62000$ shillings

Option B is to reduce books by 4, reduce sets by 2, and increase pens by 3 from Option A.
Books: $12 - 4 = 8$ books
Sets: $4 - 2 = 2$ sets
Pens: $10 + 3 = 13$ pens
Cost of Option B $= 8b + 2s + 13p$
Cost of Option B $= 8(3000) + 2(4000) + 13(1000)$
Cost of Option B $= 24000 + 8000 + 13000$
Cost of Option B $= 45000$ shillings

Budget = 60,000 shillings.
Cost of Option A = 62,000 shillings.
Cost of Option B = 45,000 shillings.
Since 62,000 shillings > 60,000 shillings, Option A is over budget.
Since 45,000 shillings < 60,000 shillings, Option B is within budget.

(i) Equations relating the costs of all items:

(ii) The cost of each item:
Book ($b$): $\boxed{\text{3000 shillings}}$
Set ($s$): $\boxed{\text{4000 shillings}}$
Pen ($p$): $\boxed{\text{1000 shillings}}$

(iii) The total cost of Option A and Option B:
Cost of Option A: $\boxed{\text{62000 shillings}}$
Cost of Option B: $\boxed{\text{45000 shillings}}$

(iv) The option he can use within 60,000 shillings budget:
$\boxed{\text{Option B}}$