![Solve the following systems of equations using the substitution method: [specific systems of equations were provided in the solution]](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1777887856008-5b4188894ad33b8f.png&w=3840&q=75)
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 systems of equations using the substitution method. 1. Given the system: x + (1)/(2)y = 5 (1) y = 4x (2) Step 1: Substitute equation (2) into equation (1). x + (1)/(2)(4x) = 5 x + 2x = 5 3x = 5 x = (5)/(3) Step 2: Substitute x=(5)/(3) into equation (2). y = 4((5)/(3)) y = (20)/(3) The solution is (x, y) = ((5)/(3), (20)/(3)). x=(5)/(3), y=(20)/(3) 2. Given the system: x = (1)/(2)y + 2 (1) x + y = 10 (2) Step 1: Substitute equation (1) into equation (2). ((1)/(2)y + 2) + y = 10 (3)/(2)y + 2 = 10 (3)/(2)y = 8 y = 8 × (2)/(3) y = (16)/(3) Step 2: Substitute y=(16)/(3) into equation (1). x = (1)/(2)((16)/(3)) + 2 x = (8)/(3) + (6)/(3) x = (14)/(3) The solution is (x, y) = ((14)/(3), (16)/(3)). x=(14)/(3), y=(16)/(3) 3. Given the system: y = (1)/(2)x - 1 (1) x + y = 7 (2) Step 1: Substitute equation (1) into equation (2). x + ((1)/(2)x - 1) = 7 (3)/(2)x - 1 = 7 (3)/(2)x = 8 x = 8 × (2)/(3) x = (16)/(3) Step 2: Substitute x=(16)/(3) into equation (1). y = (1)/(2)((16)/(3)) - 1 y = (8)/(3) - (3)/(3) y = (5)/(3) The solution is (x, y) = ((16)/(3), (5)/(3)). x=(16)/(3), y=(5)/(3) 4. Given the system: x = (2)/(5)y (1) x + y = 9 (2) Step 1: Substitute equation (1) into equation (2). (2)/(5)y + y = 9 (7)/(5)y = 9 y = 9 × (5)/(7) y = (45)/(7) Step 2: Substitute y=(45)/(7) into equation (1). x = (2)/(5)((45)/(7)) x = (18)/(7) The solution is (x, y) = ((18)/(7), (45)/(7)). x=(18)/(7), y=(45)/(7) 5. Given the system: y = (3)/(4)x (1) x + y = 14 (2) Step 1: Substitute equation (1) into equation (2). x + (3)/(4)x = 14 (7)/(4)x = 14 x = 14 × (4)/(7) x = 8 Step 2: Substitute x=8 into equation (1). y = (3)/(4)(8) y = 6 The solution is (x, y) = (8, 6). x=8, y=6 6. Given the system: x = (1)/(2)y + 3 (1) x - y = 1 (2) Step 1: Substitute equation (1) into equation (2). ((1)/(2)y + 3) - y = 1 -(1)/(2)y + 3 = 1 -(1)/(2)y = -2 y = 4 Step 2: Substitute y=4 into equation (1). x = (1)/(2)(4) + 3 x = 2 + 3 x = 5 The solution is (x, y) = (5, 4). x=5, y=4 7. Given the system: y = (2)/(3)x + 1 (1) x + y = 11 (2) Step 1: Substitute equation (1) into equation (2). x + ((2)/(3)x + 1) = 11 (5)/(3)x + 1 = 11 (5)/(3)x = 10 x =