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
Another one POPOOLA — let's solve it. To verify the given values of x and y, we will simplify both sides of the equation and then equate the real and imaginary parts. The given equation is: (2+j)/(1-j) = j(x+jy) The proposed solution is x=(3)/(2) and y=-(1)/(2). Step 1: Simplify the left-hand side (LHS) of the equation. To simplify the fraction, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 1-j is 1+j. (2+j)/(1-j) = (2+j)/(1-j) × (1+j)/(1+j) Multiply the numerators: (2+j)(1+j) = 2(1) + 2(j) + j(1) + j(j) = 2 + 2j + j + j^2 Since j^2 = -1: = 2 + 3j - 1 = 1 + 3j Multiply the denominators: (1-j)(1+j) = 1^2 - j^2 = 1 - (-1) = 1 + 1 = 2 So, the LHS simplifies to: (1+3j)/(2) = (1)/(2) + j(3)/(2) Step 2: Simplify the right-hand side (RHS) of the equation. j(x+jy) = jx + j^2y Since j^2 = -1: = jx - y = -y + jx Step 3: Equate the simplified LHS and RHS. (1)/(2) + j(3)/(2) = -y + jx For two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal. Equating the real parts: (1)/(2) = -y y = -(1)/(2) Equating the imaginary parts: (3)/(2) = x x = (3)/(2) Step 4: Compare the calculated values with the proposed values. The calculated values are x=(3)/(2) and y=-(1)/(2). These match the proposed values. ✅ Correct. The values x=(3)/(2) and y=-(1)/(2) are correct because when substituted into the equation, they satisfy the equality of the complex numbers. Send me the next one 📸