find the absolute value of z if 2z - i = 5 - 4i

To calculate the absolute value of z, we'll put z, from the
given expression, in the rectangular form.

First step is to isolate
z to the left side. For this reason, we'll add i both sides:

2z - i
+ i = 5 - 4i + i

We'll combine real parts and imaginary
parts:

2z = 5 - 3i

We'll divide by
2:

z = 2.5 - 1.5i

Now, since the
calculus of the absolute value depends on the real and imaginary parts of the complex number,
we'll identify them:

Re(z) = 2.5 and Im(z) =
-1.5

|z| = sqrt[Re(z)^2 + Im(z)^2]

|z|
= sqrt [2.5^2 + (-1.5)^2]

|z| = sqrt (15.625 +
2.25)

|z| = 4.22
approx.