(2+i)x+(1+2i)=1-4i

To solve the equation \((2+i)x + (1+2i) = 1-4i\), we will first isolate \(x\). 1. Start by moving \((1 + 2i)\) to the right side of the equation: \[ (2+i)x = (1-4i) - (1+2i) \] 2. Simplify the right side: \[ (1 - 4i) - (1 + 2i) = 1 - 4i - 1 - 2i = -6i \] So the equation now is: \[ (2+i)x = -6i \] 3. Now, solve for \(x\) by dividing both sides by \((2+i)\): \[ x = \frac{-6i}{2+i} \] 4. To divide by a complex number, multiply the numerator and the denominator by the conjugate of the denominator: \[ x = \frac{-6i(2-i)}{(2+i)(2-i)} \] 5. Calculate the denominator: \[ (2+i)(2-i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \] 6. Now calculate the numerator: \[ -6i(2-i) = -12i + 6 = 6 - 12i \] 7. Putting it all together: \[ x = \frac{6 - 12i}{5} = \frac{6}{5} - \frac{12}{5}i \] Thus, the solution is: \[ x = \frac{6}{5} - \frac{12}{5}i \]