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

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2z - i = 5 - 4i

First we wil write the expression as a complex number z:

2z - i = 5 - 4i

Let us add i to both sides"

==> 2z = 5 - 4i + i

==> 2z = 5 - 3i

Now we will divide by 2:

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

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

Now we have z in the form of z = a+ bi

Then we will calculate the absolute value:

We know that:

l z l = sqrt(a^2 + b^2)

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

=sqrt ( 25+9)/4

= sqrt34 / 2

**==> l zl = sqrt34 / 2**

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.**

The absolute value of the complex number x+yi is |x+yi| = sqrt(x^2+y^2), where x and y are real.

Therefore to find the absolute value of z in the equation 2z-i = 5-4i, we first solve for x.

2z-i = 5-4i.

2z = 5-4i +i.

2z = 5-3i

z = (5-3i).2.

z = (5/2)+(-3/2)i which is in the rectangular format like x+yi.

Therefore absolute value of z is |z| = | 5/2 +(-3/2)i| = sqrt{(5/2)^2+(-3/2)^2} = sqrt{(25+9)/4} = (1/2) sqrt(34).

Therefore |z| = (1/2)sqrt34.