Solving Equations with Complex Coefficients Simplified Revision Notes for Leaving Cert Mathematics

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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

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