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.
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Here are the solutions to the systems of equations using the elimination method. 1. Given the system: (1)/(2)x + (1)/(3)y = 4 (1) (1)/(2)x - (1)/(3)y = 2 (2) Step 1: Add equation (1) and equation (2) to eliminate y. ((1)/(2)x + (1)/(3)y) + ((1)/(2)x - (1)/(3)y) = 4 + 2 (1)/(2)x + (1)/(2)x = 6 x = 6 Step 2: Substitute x=6 into equation (1). (1)/(2)(6) + (1)/(3)y = 4 3 + (1)/(3)y = 4 (1)/(3)y = 4 - 3 (1)/(3)y = 1 y = 3 The solution is (x, y) = (6, 3). x=6, y=3 2. Given the system: (2)/(3)x + (1)/(4)y = 5 (1) (2)/(3)x - (1)/(4)y = 1 (2) Step 1: Add equation (1) and equation (2) to eliminate y. ((2)/(3)x + (1)/(4)y) + ((2)/(3)x - (1)/(4)y) = 5 + 1 (2)/(3)x + (2)/(3)x = 6 (4)/(3)x = 6 x = 6 × (3)/(4) x = (18)/(4) = (9)/(2) Step 2: Substitute x=(9)/(2) into equation (1). (2)/(3)((9)/(2)) + (1)/(4)y = 5 3 + (1)/(4)y = 5 (1)/(4)y = 5 - 3 (1)/(4)y = 2 y = 8 The solution is (x, y) = ((9)/(2), 8). x=(9)/(2), y=8 3. Given the system: (1)/(2)x + (1)/(4)y = 3 (1) (1)/(2)x - (1)/(5)y = 1 (2) Step 1: Subtract equation (2) from equation (1) to eliminate x. ((1)/(2)x + (1)/(4)y) - ((1)/(2)x - (1)/(5)y) = 3 - 1 (1)/(4)y + (1)/(5)y = 2 ((5)/(20) + (4)/(20))y = 2 (9)/(20)y = 2 y = 2 × (20)/(9) y = (40)/(9) Step 2: Substitute y=(40)/(9) into equation (1). (1)/(2)x + (1)/(4)((40)/(9)) = 3 (1)/(2)x + (10)/(9) = 3 (1)/(2)x = 3 - (10)/(9) (1)/(2)x = (27)/(9) - (10)/(9) (1)/(2)x = (17)/(9) x = 2 × (17)/(9) x = (34)/(9) The solution is (x, y) = ((34)/(9), (40)/(9)). x=(34)/(9), y=(40)/(9) 4. Given the system: (3)/(4)x + (1)/(2)y = 6 (1) (3)/(4)x - (1)/(2)y = 2 (2) Step 1: Add equation (1) and equation (2) to eliminate y. ((3)/(4)x + (1)/(2)y) + ((3)/(4)x - (1)/(2)y) = 6 + 2 (3)/(4)x + (3)/(4)x = 8 (6)/(4)x = 8 (3)/(2)x = 8 x = 8 × (2)/(3) x = (16)/(3) Step 2: Substitute x=(16)/(3) into equation (1). (3)/(4)((16)/(3)) + (1)/(2)y = 6 4 + (1)/(2)y = 6 (1)/(2)y = 6 - 4 (1)/(2)y = 2 y = 4 The solution is (x, y) = ((16)/(3), 4). x=(16)/(3), y=4 5. Given the system: (2)/(5)x + (3)/(5)y = 7 (1) (2)/(5)x - (3)/(5)y = 1 (2) Step 1: Add equation (1) and equation (2) to eliminate y. ((2)/(5)x + (3)/(5)y) + ((2)/(5)x - (3)/(5)y) = 7 + 1 (2)/(5)x + (2)/(5)x = 8 (4)/(5)x = 8 x = 8 × (5)/(4) x = 10 Step 2: Substitute x=10 into equation (1). (2)/(5)(10) + (3)/(5)y = 7 4 + (3)/(5)y = 7 (3)/(5)y = 7 - 4 (3)/(5)y = 3 y = 3 × (5)/(3) y = 5 The solution is (x, y) = (10, 5). x=10, y=5 6. Given the system: (1)/(3)x + (1)/(2)y = 5 (1) (1)/(6)x + (1)/(2)y = 4 (2) Step 1: Subtract equation (2) from equation (1) to eliminate y. ((1)/(3)x + (1)/(2)y) - ((1)/(6)x + (1)/(2)y) = 5 - 4 (1)/(3)x - (1)/(6)x = 1 ((2)/(6) - (1)/(6))x = 1 (1)/(6)x = 1 x = 6 Step 2: Substitute x=6 into equation (1). (1)/(3)(6) + (1)/(2)y = 5 2 + (1)/(2)y = 5 (1)/(2)y = 5 - 2 (1)/(2)y = 3 y = 6 The solution is (x, y) = (6, 6). x=6, y=6 7. Given the system: (3)/(5)x + (1)/(3)y = 8 (1) (3)/(5)x - (1)/(3)y = 2 (2) Step 1: Add equation (1) and equation (2) to eliminate y. ((3)/(5)x + (1)/(3)y) + ((3)/(5)x - (1)/(3)y) = 8 + 2 (3)/(5)x + (3)/(5)x = 10 (6)/(5)x = 10 x = 10 × (5)/(6) x = (50)/(6) = (25)/(3) Step 2: Substitute x=(25)/(3) into equation (1). (3)/(5)((25)/(3)) + (1)/(3)y = 8 5 + (1)/(3)y = 8 (1)/(3)y = 8 - 5 (1)/(3)y = 3 y = 9 The solution is (x, y) = ((25)/(3), 9). x=(25)/(3), y=9 8. Given the system: (1)/(4)x + (1)/(3)y = 3 (1) (1)/(4)x - (1)/(3)y = 1 (2) Step 1: Add equation (1) and equation (2) to eliminate y. ((1)/(4)x + (1)/(3)y) + ((1)/(4)x - (1)/(3)y) = 3 + 1 (1)/(4)x + (1)/(4)x = 4 (2)/(4)x = 4 (1)/(2)x = 4 x = 8 Step 2: Substitute x=8 into equation (1). (1)/(4)(8) + (1)/(3)y = 3 2 + (1)/(3)y = 3 (1)/(3)y = 3 - 2 (1)/(3)y = 1 y = 3 The solution is (x, y) = (8, 3). x=8, y=3 What's next?