For the complex number \( z = x + iy \), we seek the locus defined by \( |z - 1| = |z + 1| \). This condition signifies that the distance from \( z \) to the point \( 1 \) is equivalent to its distance to the point \( -1 \).
The distances are calculated as follows:
\(|z - 1| = \sqrt{(x - 1)^2 + y^2}\) and \(|z + 1| = \sqrt{(x + 1)^2 + y^2}\).
Equating these expressions yields:\(\sqrt{(x - 1)^2 + y^2} = \sqrt{(x + 1)^2 + y^2}\).
Squaring both sides to remove the radicals results in:\((x - 1)^2 + y^2 = (x + 1)^2 + y^2\).
Expanding both sides gives:\(x^2 - 2x + 1 + y^2 = x^2 + 2x + 1 + y^2\).
Canceling identical terms (\(x^2\), \(y^2\), \(1\)) from each side simplifies the equation to:\(-2x = 2x\).
Combining like terms produces:\(4x = 0\).
Solving for \(x\) yields:\(x = 0\).
Therefore, any complex number \(z = x + iy\) satisfying this condition must have an \(x\)-component of 0. This means \(z\) lies on the line defined by \(x = 0\), which is the imaginary axis.