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 for each system of equations using the elimination method: a) We have the system: x + 3y = 8 (1) 5x + 7y = 24 (2) Step 1: Multiply equation (1) by 5 to eliminate x. 5(x + 3y) = 5(8) 5x + 15y = 40 (3) Step 2: Subtract equation (2) from equation (3). (5x + 15y) - (5x + 7y) = 40 - 24 8y = 16 y = 2 Step 3: Substitute y = 2 into equation (1). x + 3(2) = 8 x + 6 = 8 x = 2 The solution is x=2, y=2. b) We have the system: 5x + 4y = 22 (1) 4x + 5y = 23 (2) Step 1: Multiply equation (1) by 4 and equation (2) by 5 to eliminate x. 4(5x + 4y) = 4(22) 20x + 16y = 88 (3) 5(4x + 5y) = 5(23) 20x + 25y = 115 (4) Step 2: Subtract equation (3) from equation (4). (20x + 25y) - (20x + 16y) = 115 - 88 9y = 27 y = 3 Step 3: Substitute y = 3 into equation (1). 5x + 4(3) = 22 5x + 12 = 22 5x = 10 x = 2 The solution is x=2, y=3. c) We have the system: 5x - 3y = 22 (1) 10x + 7y = 18 (2) Step 1: Multiply equation (1) by 2 to eliminate x. 2(5x - 3y) = 2(22) 10x - 6y = 44 (3) Step 2: Subtract equation (3) from equation (2). (10x + 7y) - (10x - 6y) = 18 - 44 13y = -26 y = -2 Step 3: Substitute y = -2 into equation (1). 5x - 3(-2) = 22 5x + 6 = 22 5x = 16 x = (16)/(5) The solution is x=(16)/(5), y=-2. d) We have the system: x + y = 3 (1) 3x - y = 5 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (x + y) + (3x - y) = 3 + 5 4x = 8 x = 2 Step 2: Substitute x = 2 into equation (1). 2 + y = 3 y = 1 The solution is x=2, y=1. e) We have the system: 3x + y = 5 (1) 2x - y = 0 (2) Step 1: Add equation (1) and equation (2) to eliminate y. (3x + y) + (2x - y) = 5 + 0 5x = 5 x = 1 Step 2: Substitute x = 1 into equation (2). 2(1) - y = 0 2 - y = 0 y = 2 The solution is x=1, y=2. f) We have the system: x + 4y = 6 (1) 2x + 5y = 6 (2) Step 1: Multiply equation (1) by 2 to eliminate x. 2(x + 4y) = 2(6) 2x + 8y = 12 (3) Step 2: Subtract equation (2) from equation (3). (2x + 8y) - (2x + 5y) = 12 - 6 3y = 6 y = 2 Step 3: Substitute y = 2 into equation (1). x + 4(2) = 6 x + 8 = 6 x = -2 The solution is x=-2, y=2. That's 2 down. 3 left today — send the next one.