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
Step 1: Expand the left side of the equation. Given the equation (1+2i)(x+yi) = 4+3i. Multiply the terms on the left side: (1+2i)(x+yi) = 1(x) + 1(yi) + 2i(x) + 2i(yi) = x + yi + 2xi + 2y i^2 Since i^2 = -1: = x + yi + 2xi - 2y Step 2: Group the real and imaginary parts on the left side. = (x - 2y) + (y + 2x)i Step 3: Equate the real and imaginary parts of both sides of the equation. We have (x - 2y) + (y + 2x)i = 4+3i. Equating the real parts: x - 2y = 4 (1) Equating the imaginary parts: y + 2x = 3 (2) Step 4: Solve the system of linear equations for x and y. From equation (2), express y in terms of x: y = 3 - 2x (3) Substitute this expression for y into equation (1): x - 2(3 - 2x) = 4 x - 6 + 4x = 4 5x - 6 = 4 5x = 4 + 6 5x = 10 x = (10)/(5) x = 2 Now substitute the value of x=2 back into equation (3) to find y: y = 3 - 2(2) y = 3 - 4 y = -1 Step 5: Compare the solution with the given options. The solution is x=2 and y=-1. This matches option A. The final answer is A. x=2, y=-1.