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
Here are the solutions to the problems. 2. Find the value of the unknown in the equations below: a) Given the system of equations: 3a + 4b = 32 (1) 2a + 5b = 34 (2) Step 1: Multiply equation (1) by 2 and equation (2) by 3 to eliminate a. 2 × (3a + 4b) = 2 × 32 6a + 8b = 64 (3) 3 × (2a + 5b) = 3 × 34 6a + 15b = 102 (4) Step 2: Subtract equation (3) from equation (4). (6a + 15b) - (6a + 8b) = 102 - 64 7b = 38 b = (38)/(7) Step 3: Substitute the value of b into equation (1). 3a + 4((38)/(7)) = 32 3a + (152)/(7) = 32 3a = 32 - (152)/(7) 3a = (224 - 152)/(7) 3a = (72)/(7) a = (72)/(7 × 3) a = (24)/(7) The values are a = (24)/(7), b = (38)/(7). b) Given the system of equations: 6x + 3y = 48 (1) 4x + 2y = 32 (2) Step 1: Simplify both equations by dividing by their common factors. Divide equation (1) by 3: (6x)/(3) + (3y)/(3) = (48)/(3) 2x + y = 16 (3) Divide equation (2) by 2: (4x)/(2) + (2y)/(2) = (32)/(2) 2x + y = 16 (4) Step 2: Observe that equation (3) and equation (4) are identical. This means the two original equations represent the same line. Therefore, there are infinitely many solutions. We can express one variable in terms of the other. From 2x + y = 16, we get y = 16 - 2x. The solution is Infinitely many solutions, y = 16 - 2x. c) Given the system of equations: k + 4t = 10 (1) k + 2t = 6 (2) Step 1: Subtract equation (2) from equation (1). (k + 4t) - (k + 2t) = 10 - 6 2t = 4 t = 2 Step 2: Substitute the value of t into equation (2). k + 2(2) = 6 k + 4 = 6 k = 6 - 4 k = 2 The values are k = 2, t = 2. d) Given the system of equations: 5m + 6n = 40 (1) 3m + 2n = 20 (2) Step 1: Multiply equation (2) by 3 to eliminate n. 3 × (3m + 2n) = 3 × 20 9m + 6n = 60 (3) Step 2: Subtract equation (1) from equation (3). (9m + 6n) - (5m + 6n) = 60 - 40 4m = 20 m = 5 Step 3: Substitute the value of m into equation (2). 3(5) + 2n = 20 15 + 2n = 20 2n = 20 - 15 2n = 5 n = (5)/(2) The values are m = 5, n = (5)/(2). e) Given the system of equations: 4h + 2j = 52 (1) 2h + 3j = 44 (2) Step 1: Divide equation (1) by 2 to simplify. (4h)/(2) + (2j)/(2) = (52)/(2) 2h + j = 26 (3) Step 2: From equation (3), express j in terms of h. j = 26 - 2h Step 3: Substitute this expression for j into equation (2). 2h + 3(26 - 2h) = 44 2h + 78 - 6h = 44 -4h = 44 - 78 -4h = -34 h = (-34)/(-4) h = (17)/(2) Step 4: Substitute the value of h back into the expression for j. j = 26 - 2((17)/(2)) j = 26 - 17 j = 9 The values are h = (17)/(2), j = 9. 1. A shopkeeper sold 3 kg of flour and 2 kg of sugar for sh. 180, and another sold 4 kg of flour and 3 kg of sugar for sh. 260. Find the price of 1 kg of flour and 1 kg of sugar. Step 1: Define variables. Let f be the price of 1 kg of flour (in shillings). Let s be the price of 1 kg of sugar (in shillings). Step 2: Formulate the equations based on the given information. From the first sale: 3f + 2s = 180 (1) From the second sale: 4f + 3s = 260 (2) Step 3: Multiply equation (1) by 3 and equation (2) by 2 to eliminate s. 3 × (3f + 2s) = 3 × 180 9f + 6s = 540 (3)