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

Step 1: Formulate the equations for the cost of items. 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: 4b + s + 6p = 22000 (1) 3b + 3s + 4p = 25000 (2) 5b + 2s + 3p = 26000 (3) Step 2: Solve the system of equations to find the cost of each item. From equation (1), express s in terms of b and p: s = 22000 - 4b - 6p (4) Substitute (4) into (2): 3b + 3(22000 - 4b - 6p) + 4p = 25000 3b + 66000 - 12b - 18p + 4p = 25000 -9b - 14p = 25000 - 66000 -9b - 14p = -41000 9b + 14p = 41000 (5) Substitute (4) into (3): 5b + 2(22000 - 4b - 6p) + 3p = 26000 5b + 44000 - 8b - 12p + 3p = 26000 -3b - 9p = 26000 - 44000 -3b - 9p = -18000 3b + 9p = 18000 Divide by 3: b + 3p = 6000 (6) From (6), express b in terms of p: b = 6000 - 3p (7) Substitute (7) into (5): 9(6000 - 3p) + 14p = 41000 54000 - 27p + 14p = 41000 54000 - 13p = 41000 -13p = 41000 - 54000 -13p = -13000 p = (-13000)/(-13) p = 1000 Substitute p = 1000 into (7): b = 6000 - 3(1000) b = 6000 - 3000 b = 3000 Substitute b = 3000 and p = 1000 into (4): s = 22000 - 4(3000) - 6(1000) s = 22000 - 12000 - 6000 s = 4000 The cost of each item is: Book (b): 3000 shillings Set (s): 4000 shillings Pen (p): 1000 shillings Step 3: Find the total cost of Option A. 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 Step 4: Find the total cost of Option B. 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 Step 5: Determine which option he can use within the 60,000 shillings budget. 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: 4b + s + 6p &= 22000 \\ 3b + 3s + 4p &= 25000 \\ 5b + 2s + 3p &= 26000 (ii) The cost of each item: Book (b): 3000 shillings Set (s): 4000 shillings Pen (p): 1000 shillings (iii) The total cost of Option A and Option B: Cost of Option A: 62000 shillings Cost of Option B: 45000 shillings (iv) The option he can use within 60,000 shillings budget: Option B Send me the next one 📸