1. **State the problem:** Solve the simultaneous linear equations involving complex numbers. 2. **General approach:** For simultaneous linear equations, we use substitution or elimination methods. When complex numbers are involved, treat the real and imaginary parts separately or solve algebraically as usual. 3. **Example problem:** Suppose the system is: $$\begin{cases} a + bi = c + di \\ e + fi = g + hi \end{cases}$$ where $a,b,c,d,e,f,g,h$ are real numbers and $i$ is the imaginary unit with $i^2 = -1$. 4. **Step-by-step solution:** - Separate real and imaginary parts from each equation. - Equate real parts and imaginary parts separately to form a system of real equations. - Solve the resulting real system using substitution or elimination. 5. **Important rules:** - $i^2 = -1$ - Complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. 6. **Example:** Solve $$\begin{cases} x + yi = 3 + 4i \\ 2x - y i = 1 + 2i \end{cases}$$ Separate real and imaginary parts: - From first equation: real $x = 3$, imaginary $y = 4$ - From second equation: real $2x = 1$, imaginary $-y = 2$ From real parts: $2x = 1 \Rightarrow x = \frac{1}{2}$ From imaginary parts: $-y = 2 \Rightarrow y = -2$ 7. **Check for consistency:** The first equation gave $x=3$, $y=4$, second gave $x=\frac{1}{2}$, $y=-2$, so no solution unless equations are consistent. 8. **Conclusion:** For consistent systems, solve real and imaginary parts separately. If inconsistent, no solution. **Final answer:** The solution depends on the specific equations given. Use the method above to solve any simultaneous linear equations with complex numbers.