Linear Complex Equations

Principle

A linear complex equation is a first-degree equation for an unknown complex number. It has the same algebraic shape as a real linear equation, but the coefficients and the solution may be complex numbers.

Because every non-zero complex number has a reciprocal, a linear complex equation with \(a\ne0\) has exactly one complex solution.

Notation

• \(z\) is the unknown complex number.
• \(a\) is the coefficient of \(z\).
• \(b\) is the constant term.
• \(\overline{a}\) is the complex conjugate of \(a\).
• \(|a|\) is the modulus of \(a\).

Method

Start with the equation:

Subtract \(b\) from both sides:

Divide by \(a\), which is allowed because \(a\ne0\):

If the denominator is complex, write the quotient in standard form by multiplying top and bottom by the conjugate of the denominator:

The denominator \(|a|^2\) is real and positive, so the final expression can be separated into real and imaginary parts.

Rules

• A degree-one complex equation has one solution when the coefficient of \(z\) is not zero.
• If \(a=0\) and \(b\ne0\), the equation has no solution.
• If \(a=0\) and \(b=0\), every complex number is a solution.
• Complex numbers are not ordered, so solving does not involve choosing a larger or smaller solution.

Examples

Solve:

First isolate \(z\):

Divide by \(2-i\):

Multiply top and bottom by \(2+i\):

Compute the numerator and denominator:

Therefore:

Checks

• Check that the coefficient of \(z\) is not zero before dividing.
• Use the conjugate of the denominator, not the numerator, when simplifying a complex quotient.
• Substitute the result back into the original equation to catch sign errors.
• Keep the final answer in standard form \(x+iy\) unless polar form is requested.