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
...(16xy)/(xy) (1)/(y) + (2)/(x) = 16 (4) Step 2: Let a = (1)/(x) and b = (1)/(y). Substitute these into equations (3) and (4) to form a linear system. The system becomes: 2b - a = 2 (3') b + 2a = 16 (4') Step 3: Solve the new system for a and b. From equation (3'), we can express a in terms of b: a = 2b - 2 Substitute this expression for a into equation (4'): b + 2(2b - 2) = 16 b + 4b - 4 = 16 5b - 4 = 16 5b = 20 b = 4 Now substitute the value of b back into the expression for a: a = 2(4) - 2 a = 8 - 2 a = 6 Step 4: Find x and y using a = (1)/(x) and b = (1)/(y). Since a = 6, we have (1)/(x) = 6, which means x = (1)/(6). Since b = 4, we have (1)/(y) = 4, which means y = (1)/(4). The solution to the system is: x = (1)/(6), y = (1)/(4) We can verify these values in the original equations. For 2x - y = 2xy: 2((1)/(6)) - (1)/(4) = (1)/(3) - (1)/(4) = (4-3)/(12) = (1)/(12). And 2((1)/(6))((1)/(4)) = 2((1)/(24)) = (1)/(12). This is correct. For x + 2y = 16xy: (1)/(6) + 2((1)/(4)) = (1)/(6) + (1)/(2) = (1+3)/(6) = (4)/(6) = (2)/(3). And 16((1)/(6))((1)/(4)) = 16((1)/(24)) = (16)/(24) = (2)/(3). This is also correct. The final answer is x = (1)/(6), y = (1)/(4). That's 2 down. 3 left today — send the next one.