This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) Given matrix P = 7 & 9 \\ 11 & 8 . To find the transpose of matrix P, denoted as P^T, we swap its rows and columns. P^T = 7 & 11 \\ 9 & 8 The transpose of matrix P is 7 & 11 \\ 9 & 8 . b) Let e be the price of one exercise book and g be the price of one graph book. From the first statement: Nesta bought 7 exercise books and 9 graph books for Kshs 890. 7e + 9g = 890 (1) From the second statement: If she bought 11 exercise books and 8 graph books, she would pay Kshs 140 more than 890. 11e + 8g = 890 + 140 11e + 8g = 1030 (2) Step 1: Write the system of equations in matrix form M e \\ g = C. 7 & 9 \\ 11 & 8 e \\ g = 890 \\ 1030 Step 2: Find the determinant of the coefficient matrix M = 7 & 9 \\ 11 & 8 . (M) = (7)(8) - (9)(11) = 56 - 99 = -43 Step 3: Find the inverse of the coefficient matrix M^-1. M^-1 = (1)/((M)) 8 & -9 \\ -11 & 7 = (1)/(-43) 8 & -9 \\ -11 & 7 Step 4: Multiply M^-1 by the constant matrix C to find e \\ g . e \\ g = (1)/(-43) 8 & -9 \\ -11 & 7 890 \\ 1030 e \\ g = (1)/(-43) (8)(890) + (-9)(1030) \\ (-11)(890) + (7)(1030) e \\ g = (1)/(-43) 7120 - 9270 \\ -9790 + 7210 e \\ g = (1)/(-43) -2150 \\ -2580 e = (-2150)/(-43) = 50 g = (-2580)/(-43) = 60 The price of one exercise book is Kshs. 50 and the price of one graph book is Kshs. 60. c) Step 1: Calculate the new price of an exercise book at shop B. The price of an exercise book at shop B is 18% cheaper than in shop A. Price of exercise book in shop A (e_A) = Kshs 50. Price of exercise book in shop B (e_B) = e_A × (1 - 0.18) = 50 × 0.82 = 41 Kshs. Step 2: Calculate the new price of a graph book at shop B. The price of a graph book at shop B is 5% higher than in shop A. Price of graph book in shop A (g_A) = Kshs 60. Price of graph book in shop B (g_B) = g_A × (1 + 0.05) = 60 × 1.05 = 63 Kshs. Step 3: Calculate the total cost if Nesta bought the items from shop B. Nesta bought 7 exercise books and 9 graph books. Cost at shop B = (7 × e_B) + (9 × g_B) Cost at shop B = (7 × 41) + (9 × 63) Cost at shop B = 287 + 567 = 854 Kshs. Step 4: Calculate the amount of money she would have saved. Original cost at shop A = Kshs 890. Amount saved = Original cost at shop A - Cost at shop B Amount saved = 890 - 854 = 36 Kshs. The amount of money she would have saved is Kshs. 36. That's 3 down. 2 left today — send the next one.