Solving Equations with Complex Coefficients Revision Notes for Leaving Cert Mathematics

Solving Equations with Complex Coefficients

When solving equations with complex coefficients, we first need to know how equality works under the complex system :

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Consider two complex numbers : $z=a+bi$ and $w=c+di$ . $z=w$ if :

Re = Re (reals components are equals) and Im = Im (imaginary components are equal).

Example

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Solve for $x$ and $y$ in the following equation :

4x-3i(x+2y)=y+9

First identify all components of the equation :

\underbrace{4x}_\text{Re}-3i(x+2y)=\underbrace{y+9}_\text{Re}

Equate real components with real components :

4x=y+9

Identify imaginary components :

4x\underbrace{-3i(x+2y)}_\text{Im}=y+9+\underbrace{0i}_\text{Im}

Equate imaginary components (discard the $i$ , since all terms have this anyway)

-3(x+2y)=0 \\ -3x-6y=0

We have now derived two equations which we can solve simultaneously :

\begin{align*} 4x-y=9 \\ -3x-6y = 0 \end{align*}

\begin{align*} -24x+6y=-54 \\ \underline{-3x-6y = 0} \\ -27x=-54 \end{align*}

\begin{align*} x=2 \end{align*}

4(2)-y=9

y=-1