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How to Solve Linear Equations with Complex Numbers

You can solve linear equations involving complex numbers by using the same techniques as you use when solving linear equations involving real numbers. In order to solve linear equations that have complex numbers, it is essential that you’ve mastered addition, subtraction, multiplication and division using complex numbers.

Example 1

Solve the equation $10 z - 8 i = 1 - 5 i z$ for $z$

You start by moving all terms involving $z$ to one side of the equal sign:

\begin{array}{l} 10 z - 8 i & = 1 - 5 i z, \\ 10 z + 5 i z & = 1 + 8 i, \\ (10 + 5 i) z & = 1 + 8 i . \end{array}

Then you isolate $z$ by dividing by the coefficient:

\begin{array}{l} (10 + 5 i) z & = 1 + 8 i, \\ z & = \frac{1 + 8 i}{10 + 5 i} . \end{array}

You always want a solution that doesn’t involve a complex number in the denominator. Thus you need to expand the fraction by the complex conjugate of the denominator:

\begin{array}{l} z & = \frac{1 + 8 i}{10 + 5 i} \\ = \frac{(1 + 8 i) (10 - 5 i)}{(10 + 5 i) (10 - 5 i)} \\ = \frac{50 + 75 i}{125} \\ = \frac{2 + 3 i}{5} . \end{array}

Example 2

Solve the equation $\frac{z - 3}{z + i} = i$ for $z$

In order to solve the equation, you must first multiply both sides of the equation by $z + i$ : $\begin{array}{l} \frac{z - 3}{z + i} & = i, \\ z - 3 & = i z - 1 . \end{array}$

You can now solve the equation by moving all terms involving $z$ to one side of the equal sign:

\begin{array}{l} z - 3 & = i z - 1, \\ z - i z & = 2, \\ (1 - i) z & = 2 . \end{array}

Isolate $z$ by dividing both sides of the equation by $1 - i$ : $\begin{array}{l} (1 - i) z & = 2, \\ z & = \frac{2}{1 - i} . \end{array}$

The final answer is obtained by expanding the fraction by the complex conjugate of the denominator:

\begin{array}{l} z & = \frac{2}{1 - i} \\ = \frac{2 (1 + i)}{(1 - i) (1 + i)} \\ = \frac{2 (1 + i)}{2} \\ = 1 + i . \end{array}

Think About This

Linear equations with real numbers can be solved by graphing each side of the equal sign and finding the point of intersection. Will this work for linear equations with complex numbers?

Real-valued functions can be plotted in a two-dimensional coordinate system. On the other hand, plotting complex functions requires four dimensions. Since drawing four dimensional coordinate systems is impossible, plotting complex functions is also impossible. Thus, equations involving complex numbers can’t be solved by graphing.

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