Find the absolute value of zFind the absolute value of z if   2z - i = 5 - 4i.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have 2z - i = 5 - 4i

2z - i = 5 - 4i

=> 2z = 5 - 3i

z = (5/2) - (3/2)i

|z| = sqrt ((5/2)^2 + (3/2)^2)

=> (1/2)sqrt(25 + 9)

=> (sqrt 34)/2

The absolute value of z is (sqrt 34)/2

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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